(This pertains to U.S. universities.) Times may have changed, but when I was in graduate school (several places, and yes I know this is unusual, but I mention it because I'm talking about more than one data point) this was a standard topic that showed up somewhere in the standard first year (2-semester) graduate algebra sequence (e.g. Lang, Hungerford, Jacobson, etc.), and the only way an undergraduate would see this in a formal class would be when taking such a class as an undergraduate. Of course, someone also taking upper level physics classes might see physics/engineering tensors in classical mechanics, continuum mechanics, fluid dynamics, electrodynamics, etc. (in my case it was in the form of dyads that appeared in the second semester of a 2-semester sequence using Symon), and someone taking an honors level advanced calculus course at one of the few universities that offered such courses (think Loomis/Sternberg, Nickerson/Spencer/Steenrod, or Fleming) would see tensor products, but in these cases (and probably others I could imagine) we're either not talking about formal tensor products in which notions such as "universal mapping property" arise or we're not talking about standard undergraduate material. I checked my undergraduate linear algebra text, Hoffman/Kunze, which was the text used for a fairly stiff undergraduate course in linear algebra that I took in 1978, and I see some discussion of tensor products in Sections 5.6: Multilinear Functions and 5.7: The Grassman Ring (pp. 166-180), but in looking over this now I definitely remember that we skipped these sections (further evidence is that I also have margin notes and homework problems marked in Section 5.4, then none until Chapters 6 and 7 and the beginning of Chapter 8).