There isn't any problem with that. In Euclidean Geometry, the parallel axiom is assumed, which, simply stated, means that given a line and a point not on that line in the plane, there is only one line that passes through that point, and does not cross the original line. Previous assumptions can prove that at least one parallel exists.
Now, in Hyperbolic Geometry, the exact opposite is assumed: that there are more or less than one parallel line, which, because we already know that there is at least one, means that there is at least another parallel line.

By the way: even the most basic axiom in Euclidean Geometry is an assumption of existence: "between every two points there exists one line."
In Algebra, one of the axioms of groups is that a group has a neutral element.
The "Axiom of Choice" is an existence axiom: "given a set of non-empty sets, there exists a set that contains one element out of each set."

If we move towards theorems, Analysis is notorious for "existence theorems," namely, theorems that state the existance of certain objects, without pointing them out explicitly. An example:
"Let f be a continuous function on the interval [a,b], and f(a) < f(b). Then for every x in (f(a),f(b)) there exists a point c in (a,b) such that f(c) = x."

The axiom I presented for Hyperbolic Geometry is the correct one. It can be used to prove that there are infinitely many parallel lines to each line, going through a point not on it.

But you are correct, my statements were a bit irrelevant. And your last conjecture seems to be true, to me. Mathematics, as far as I know it, is not really interested in what exists in the real world, and what doesn't. It does interest itself in the logical consequences of properties of objects. An Applied Mathematician can then say "Lo, for this Physical/Engineering-related/Literary(?) problem can be translated into yon Mathematical object, where it can be dealt with by the tools of Mathematics."