Tip: Companion pets are account wide. If one of your characters has a pet all your characters can use that pet.
Not if an alt is the other faction
This is only the case for a rare few pets that are faction specific. The tool tip in the pets menu usually states which ones those are - if it is red it will tell you why. Also, most of the ones that are faction specific have a version for both Horde and Alliance. If you switch factions, the other one becomes available.