Yes, but it goes beyond that, because it's more difficult to answer the question of whether a mathematical structure "objectively exists", and it's not just about the question of the existence of, say, numbers.

Consider axiomatic systems, one example of which is set theory, which is very basic to modern mathematics. The particular axioms you choose affect how it behaves - for example, there's ZF set theory and ZFC set theory, which are distinguished by whether or not they include the axiom of choice. There are also alternatives to set theory that take a different approach, such as category theory.

Aliens are quite likely to come up with something different yet again. At some level, it tends to be possible to map these systems to each other, and such correspondences are often considered significant because they suggest something more fundamental than the theories themselves. But we can't directly use those fundamental correspondences without making up some sort of system with which to talk about and manipulate them, and the choices we make affect what manipulations we can do.

As such, the mathematics we actually use - as opposed to the Platonic ideal we might hold in our imaginations, what Erdos called "The Book" - is a social construction of humans, that is constrained in certain ways by rules that we choose to include in the systems we construct.