The title of your question presents an interesting dichotomy. It is of course true that a necessary condition to succeed as a mathematician is that one be "strong at mathematics"...but exactly what that means is less clear-cut. And whatever "strong at mathematics" means, it is something that can be developed (or not) over time: it is not something that either inheres in one's soul or doesn't like some sort of Calvinist election.
This may be biological, for I was psychologically tested and found that my linguistic skills were abnormally high but my spatial reasoning...wasn't (I was diagnosed with Asperger's Syndrome).
Wondering as a teenager what one may or may not be "biologically capable of" is not something that I would recommend: you are placing too much stock in biology and testing. (Frankly, I am skeptical even of the Asperger's diagnosis. I am by no means professionally trained in psychology, but in my opinion there is a faddishness in the diagnosis of such "syndromes". I believe that Asperger's diagnoses are based on measurable cognitive differences, but worry that the implications of these differences may be exaggerated. To quote from the wikipedia article on Asperger's: "Some researchers have argued that AS can be viewed as a different cognitive style, not a disorder or a disability, and that it should be removed from the standard Diagnostic and Statistical Manual, much as homosexuality was removed.") Also there is much, much more to mathematics than "spatial reasoning": I am a research mathematician with some kind of international reputation, but my inherent spatial reasoning abilities are no better than average. I bought a bookcase a few months ago, but delayed assembling it and seem to have lost the instructions: without them, I think I would rather buy a new bookcase than worry about how to assemble the one I already have! When it comes to applying spatial reasoning to mathematics, I can see that I have to work harder than many of my peers but I can still do it: for instance, when I discuss finding volumes of revolution in calculus I can always identify a few students who can visualize the regions more easily than I can...but overall I am still better at these problems than almost any calculus student I have ever taught because (i) I have so much more experience with these problems than they do and (ii) there are aspects of solving these problems other than spatial reasoning. I think that if you want to work in certain aspects of low-dimensional topology then it is important to have strong spatial reasoning skills -- sometimes I have seen talks or read papers in which proofs in these areas are done with the aid of pictures that look very confusing to me -- but even here I think that other skills are just as important or more as naked spatial reasoning. (In fact I did undergraduate research in low-dimensional topology, really enjoyed it, and would be happy to do more of it some day.)
I do not want this to deter me from pursuing a degree in physics/math and finding work in the field.
It is good to keep an eye on the future, but if you try to deal with the future as if it were the present then you're bringing a lot of unnecessary stress on yourself....we simply don't know so well what the future will bring. Having a goal of getting an undergraduate degree in physics or math is a good level of future planning for a high school sophomore. You can also start looking into the various careers that make use of this degree: there are many, and most of them would not go under the name "mathematician". The best way to stay on track towards this goal is simply to take challenging math classes, do your best to get what you can out of them, and also feed your interest in mathematics by reading up on whatever outside of class mathematical material interests you.
On the other hand, what if I just don't have the ability and fail?
We all worry about that. I still worry about that (maybe my definitions of "ability" and "failure" have been adjusted, but the emotional effect is much the same). I feel very confident that the world's leading mathematicians have these thoughts as well. They're helpful up to a point -- the point where they drive us to improve ourselves and have lofty goals -- and then they stop being helpful (at the point where we uselessly worry about our innate abilities or future success rather than try to figure out what to do next).
Recently, I took a math test and felt completely lost, even though I studied diligently and received high scores on other assignments, which may be due to the anxiety I felt as I received the exam. The mere thought of that test kills me, as if it is a reminder that I shall never succeed in the subject no matter how hard I try (I was given a B last semester).
It happens -- none of this is proof positive that you are not cut out for a mathematical career. When I was in school I usually did very well in my math classes but not always: I remember one "honors precalculus" course I took: the teacher was just very intense and had a way of making things tricky and complicated. I think he was probably a very good teacher, but his high ambitions and intensity did not translate well for me: I remember when we started a unit on linear equations, a subject which I had seen in several previous courses and knew was easy to understand. But somehow he made that material complicated as well: he discussed at least four (!!) different forms of equations of lines, and the exam managed to contain questions on linear equations that were really difficult! For one of the four quarters I got a $B$ in this course, and it was a bit discouraging. The next year I took calculus and it was both easier and more interesting than the precalculus course I had taken the year before: but I remember that though I found that most everything in the calculus course came very quickly, I also spent more than two hours a night solving calculus exercises...which must have been much more time than I had put into the "harder" precalculus course.
I was wondering if those in advanced mathematics have had the same experiences, and somehow overcame their struggles
In the first graduate level mathematics course I took, I studied "the wrong version of the Radon-Nikodym Theorem" for the final exam and was unable to do anything with one out of the three questions. I got a $B$ in that course and later found out that there had been some real question as to whether I should be allowed to continue with the graduate analysis sequence. In the end I got an award for being one of the best undergraduate math majors. Being "strong at mathematics" doesn't mean that you are inhumanly perfect: virtually everyone has at least a few lack-of-success stories to tell.
Is it optimal I should just study a humanities subject even though this is what I like, or is there some way I could find a path in the field? Also, as an aside, what gives you your passion for math? What is the best thing you find about this subject? Perhaps that can force my motivation to the point where I become proficient...
What I find interesting about your question is that while you sound very confident about your skills in language and the humanities, you seem very intent on pursuing a mathematical field....but you don't really say why you like mathematics better than humanities. (I'm not blaming you: such things can be hard to explain.) I can identify with this, because all throughout my pre-collegiate education I had skills in the humanities which were as strong as those I had in mathematics and more consistent: I never had a day where e.g. I tried to read Shakespeare and failed! Perhaps I had the sense that mathematics was deeper and more challenging than the humanities and was drawn to the idea that there was more clear room for improvement. As a high school student I learned a bit about calculus and saw that there were interesting things that lay far beyond -- like number theory, which I have been interested in since my sophomore year of high school -- and I'm not sure what is the equivalent in the humanities of "gunning for number theory". On the other hand I took creative writing as a high school sophomore, enjoyed that immensely and felt approximately the same infinite challenge as I did (and still do) with mathematics. I am somewhat regretful that I gave that up...although, given that I have more than 2000 pages of "extra" mathematical writing on my webpage, perhaps I have not really given it up so completely. So I wonder:
What is the right path for a high school student who is extremely talented and ambitious in the humanities?
I didn't know the answer to that question then, and I still don't now...but I wish I did. You might want to look into it. If you feel much more confident in your abilities in the humanities, shouldn't you at least think about going into the humanities?
Let me also say that the phrase about "forcing my motivation until I become proficient" worries me a bit. You also ask what gives mathematicians their passion for math and what they like about it. To me this sounds a little like someone who is thinking of becoming engaged asking an older married friend exactly what it was about her spouse that made her decide to get married. If you have to ask, then maybe you should be dating someone else! My passion for mathematics comes from the fact that I love it...it is not really something that can be further analyzed or explained. If you love mathematics, spend more time with it and develop your knowledge and skills. Otherwise keep taking mathematics courses at least until you get to college, but keep your mind open to finding the true object of your affections. It will be out there somewhere...