Sure; I think we’re in violent agreement here, it’s absolutely the case that people write the simpler version when the meaning is clear from context. I’ve definitely done that a bunch.
In my current ring theory course, we have indeed written things like 3 * 3 = 1 when working in |F_5 (not sure that notation is going to work as well as I hope, looks alright in the app I use), but it's not the equality symbol is overloaded, but the numbers themselves. Rather than using = to mean numeric equality and equality w.r.t. equivalence classes, we just use the numbers themselves as shorthand for their equivalence classes.
This is not limited to rings of the form Z/nZ.