What is the difference in a bass compressor and a vocal compressor? Are they all the same? Can you use any kind for bass?

Also, if my PA has a compressor is it necessary to have one in my bass rig as well?
Most compressors are the same and I wouldn't draw a distinction in most rack gear. Some instrument specific compressors purposefully color your tone quite a bit while most vocal compressors are as transparent as possible.

A lot of vocal compressors also come with processing to remove sibilance or might have a little bit of eq built in that emphasizes vocal frequency ranges. Just depends.

It's not necessary to have a compressor in your bass rig at all. If you have a need for one, then get it, if you don't, then don't bother. You could have a compressor both in your bass rig and at the PA, but that just depends on your needs.