How does one become a mathematician? I mean how someone goes from a student to become a teacher or a professor ? What is that state feel like ? Are you confident of the knowledge you have conquered over the years ?or there is part of doubt in there ?
I am current in the 3rd year out of 4 to get my degree, and I am struggling to learn the subjects to the fullest, always questions about things that confuse me and  I can't answer , I can't solve problems with confidence (a lot of times I can't solve them at all), I don't have much confidence about my solution etc. I have a feeling that even if I get my degree those doubts and confusion  won't go away like that, if there is someone who felt that way their opinion and point of view are more that welcome . You can imagine you are giving advice to your old self , an advise you never got but you needed.
For example because you did good on your finals doesn't mean you know abstract algebra , it's like the piano, you can't call your self a pianist because you know how to play well 5-10 songs. So when someone can be called a mathematician ? Or even that at least he knows abstract  algebra. Also,  I can observe that my knowledge on statistics and probabilities are a bit weaker over the pure math like real analysis number theory etc(because i study them more and give them more attention). Does every professor know deeply all the subjects ?
 A: Given that I don't know you, nor does anyone else on this forum probably, I cannot say whether your doubts are justified or not. I can only share my limited experience.
Almost my entire (short) mathematical career was centered around algebraic number theory. I've taught a few courses in abstract algebra over the years, from group theory to algebraic geometry. My knowledge of statistics and real analysis is extremely limited, somewhere around second-year level. Certainly nowhere near research level, I have no deep understanding of anything related to those subjects.
I started to gain confidence in my ability as a mathematician once I became a teaching assistant, when I got to see the work of dozens and dozens of other students. Seeing their perspectives, their approaches, and hearing their questions greatly broadened my own mathematical toolbox, and it also made clear to me that my mathematical abilities were at least above average.
At first, teaching (tutorials, as a TA) was quite frightening and I did not think I was fit for the job. After a few months of weekly tutorials I was able to answer most questions quite comfortably, and deliver coherent and somewhat interactive lectures (of 20-30 minutes). After about a year I was comfortable discussing problems instead of simply answering them, and comfortable admitting that I didn't know an answer, and maybe even investigating the question together with the students. Around this time I started considering myself to be an actual teacher.
