Correction on how to phrase your question - Does... To be out of bounds” and to be off limits” have the same meaning? Yes they mean the same thing.

Your question was posed correctly. Do a and b have the same meaning? Does a have the same meaning as b? 'out of bounds' comes from sports. The ball went out of bounds off of Messi's head. (not 'off limits') The third floor is off limits to students. (not 'out of bounds') (not permitted) Personal questions are out of bounds/ off limits in a job interview. (both work here and the meanings are similar)

Yeah, they have similar meaning.