Meaning of 'argument of a function' What is a function's argument? I often see '$f(x,y)$ has two arguments $x$ and $y$, Is having two arguments either the fact that the function can take two numbers, or does it have two specific variables which are tied to it? It seems wrong to talk this way, as I can easily have $f(x)$ or $f(z)$ and that doesn't imply $x=z$.
Is, 'in the expression $f(x,y)$ $x$ and $y$ are the arguments (the number $x$ and the number $y$ are acting as the inputs).

We can start from a natural language example, the expression "The capital of France is Paris".
In it we have two names: "Paris" and "France". If we replace into this expression the name "France" with a different name, e.g. "Italy" what we get is a different expression with a different meaning. And the same if we replace the name "Paris" with "Rome".
Having performed this "operation", we may imagine that the expression is composed of a stable part: "The capital of... is___", expressing the relation between two objects (denoted by the names) standing in this relation.
We can describe this fact in an abstract way saying that the first component is a function and the latter its arguments.
Freely derived from G.Frege's Begriffsschrift (1879).
This is the ubiquitous use of functions in mathematics: the sine and cosine functions are trigonometric functions that, for a specified angle, express the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse) and the ratio of the length of the adjacent leg to that of the hypotenuse, respectively.
For an angle $\theta$, the sine and cosine functions are denoted simply as $\sin \theta$  and $\cos \theta$, where to be precise, $\sin$ and $\cos$ are the names of the two functions and $\theta$ is the name of the argument.
See also the post Concept of a function and idea of a formula as a function for the modern origin of the mathematical concept of function as a sort of "rule" [expressed symbolically] that, having received a value as “input” allows us to calculate a corresponding “output” value.

Regarding your misunderstanding: "I can easily have $f(x)$ or $f(z)$ and that doesn't imply $x=z$", you have to try to understand the abstract concept of function, that we can express in many ways.
A function, see the natural language above, can be described with the expression "Blah blah ..." where the dots denotes the argument place, i.e. the empty place to be filled with the name of the input value.
The customary usage is $f(x)$ but we can as well use $f( \ )$ or $f(\_)$. But we may have problems with functions of more than one argument, where we must write something like $f(\circ, \square)$ or $f(\__{1},\__{2})$.
Common practice is $f(x,y)$.
