Imagine you are in inside of a giant beach ball, walking on the surface. You will never find an edge, and it is meaningless to talk about such a thing. Yet the space is finite.

A beach ball has no "edge", but it definitely has a boundary surface, and everything on the other side of this surface is "not beach ball".

Now what lies beyond the universe's boundary surface? Is such a surface even present?

That is indeed where the analogy starts to crumble. The reason is that space itself is curved. It is curved by the matter inside it. Still the property holds that if you travelled far enough fast enough you’d come back to where you started.

Conversely, the surface of the beach ball has no "center". It's also fun to realize if you blow that beach ball up, from the point of view of any point on the ball, all other points grow further away.

The universe may or may not be infinite, for some particular definition of infinity. Our light cone, which essentially defines our universe from our point of view, is distinct and measurably finite.

I don't think my physician calls the edge of the universe anything unless he's a hobbyist physicist as well.

It may mean beyond the observable universe.