course names are not universal; as you indicate you university isn't teaching what you expected in other places, so why should these be any different?
in order to stfy differential geometry you should have a sound knowledge of the basicas of analysis (limits, differentiation of vector fucntions, integration), vector spaces (say to the point where you understand that the dual space of a finite dimensional vector space is isomorphic [but unnaturally] to its dual space), and topology with perticular regard to metric spaces.
for instance, a differential manifold is a topological space with an atlas, that is a collection of open subset U_a, for a in A (possibly a finite index , possibly infinite) such that each U_a is homeomorphic to a nieghbourhood of the origin R^n and such that it is well 'behaved on intesections' ot things like (U_a)n(U_b) {i don't think it helps to talk too much about this. we then can transfer the analytic properties of these nbds in R^n to the U_a by pullnig back via these maps.
i would be surprised if anywhere in the US taught this before the 3/4th year of undergraduacy if at all. In the Uk I'd have it as a third year course, possibly a second year one depending on how much detail they did it in. eslewhere in europe it could well be in the second year.
