Difference between arcsin and inverse sine. I first learned that arcsin and inverse sine are two ways of saying the same thing.
But then I was thinking about the inverse sine function being a function, so it must be limited in it's range from -1 to +1.
What would you call the sine function reflected through the line y=x that is not limited in range ? 
Sure, it would not be a function but what would you call this graph ?
Could it be that arcsin is not a function and has infinite solutions whereas inverse sine is a function and has only one solution, e.g. $\arcsin (0.5) = \frac{\pi}{6}+2n\pi,n \in\mathbb Z, \sin^{-1}(0.5)=\frac{\pi}{6}$ ?
If not and they both have only one solution then how would you express the graph that has infinite solutions ?
I found this:
https://www.dummies.com/education/math/trigonometry/how-to-distinguish-between-trigonometry-functions-and-relations/
It seems a convention of capital/small letters are used to distinguish functions and relations.
 A: these are in fact the same function.  $\arcsin$ is the better notation I feel, because many students think that they are equal $$\sin^{-1}(x) \neq \frac{1}{\sin(x)}$$
when in fact $\arcsin(x)$ (or $\sin^{-1}(x)$) is the unique function so that:
$$\arcsin(\sin(x)) = \sin(\arcsin(x)) = x $$
Now to address your other question.  $\arcsin(x)$ IS a function.  If you wish to know it's domain and range, first consider the domain and range of $\sin(x)$
This is a bounded periodic function whose domain is all of the real numbers and whose range is from -1 to 1 inclusive.  Recall that the domain and range of inverses switch.  Therefore the domain of $\arcsin$ is from -1 to 1 with vertical asymptotes at -1 and 1 and the range is all real numbers:

A: If I understand your question correctly, you want to ask the following:

  
*
  
*I have a function which is not injective; this means that there exists two values $x_1$ and $x_2$ with $x_1\neq x_2$ and $f(x_1)=f(x_2)$
In your case: $f(x)=\sin(x)$
  
*Now I'm drawing the graph $x=f(y)$
Because the function is not injective the graph contains multiple points with the same $x$ coordinate.
  
*Does a certain term (word, name) exist for such kinds of graphs?
  
*Is there a special term for some kind of function $g(y)=\{x:f(x)=y\}=\{x_1,x_2,...\}$?
  

I doubt that there is a certain term (word, name) for such graphs and/or functions in general.
Especially in the case of the "inverse function" you'll have to keep in mind that a function has exactly one function value for a given function argument.
This means that: $f(x_1)=f(x_2)\Rightarrow g(f(x_1))=g(f(x_2))$.
It is not possible to define a function with $\sin^*(0)=0$ and $\sin^*(0)=\pi$ the same time!
You could however define a function whose function value is a set rather then a number.
Example:
$f(x)=x+5\\
f^{-1}(y)=y-5\\
f^*(y)=\{y-5\}$
Note that $f^{-1}(y)\neq f^*(y)$ because $f^{-1}(y)$ is a number and $f^*(y)$ is a set!
You might now define $sin^*(y)=\{x:sin(x)=y\}$ so $sin^*(0)=\{0,\pm\pi,\pm 2\pi,\pm 3\pi...\}$.
However I doubt that a name of such a function exists.
