Is it possible to tell if two sequences of assembly instructions are semantically equivalent?
If you restrict to programs of some fixed, finite length then yes, this is decidable. There are only a finite number of possible programs so there is a finite list of which ones are equivalent to one another. A program that is hard-coded with that list can answer the question just by looking up the two programs.

However, if you don't restrict the length of the programs, the problem becomes undecidable: if it were decidable, you could solve the halting problem ("Does program&nbsp;$P$ halt with input&nbsp;$I$?") by rephrasing it as "Is the program that simulates $P$&nbsp;running on input&nbsp;$I$ equivalent to this trivial program that always halts?"

As an aside, why do you feel that a compiler would need to do this? Compilers just take a program represented in one form (source code) and translate it into another form (the executable image). The translation process guarantees that the two are semantically equivalent, so there's no need to check.