Take a square (without border) and build two bridges on it. You can go under the bridge or across the bridge (as my attempt at drawing it poorly attempts to describe).
Since you can freely move the bridges around, on one hand I would say this surface is the same as a torus: just one bridge would turn it into a cylinder (connecting two sides) and then putting another bridge on it turns it into a mug, which is a torus. (also orientation is ^ ^, > >)
On the other hand the square with just one bridge would seem to have two different "generators" (i.e. not homotopic to a point) already, the two white ones I drew (imagine them on the same bridge). But then the square with two bridges would have four... which means it is not a torus.
I am confused: what is the type of this surface?
The right white path you drew is homotopically trivial. It lives exclusively on the square. Your entire surface is homotopic to the number symbol '8' if you shrink the square down to 2 paths connecting the bridge ends.
So the homotopy group is a free product of two copies of $\mathbb{Z}$, for a torus it is the direct sum of two copies of $\mathbb{Z}$.