@sebbabl I'm not entirely sure the total number across all legends and it would be hard for me to count them out exactly because there have been a ton of event exclusive cosmetics. There have also been the BP exclusive content as well.
How ever to answer your question there were initially 8 cosmetics. How ever there have been additions, Dive Emotes, Holos, and weapon charms. So I believe there are now 11 different types of cosmetics. If you count heirlooms than there are 12 different types of cosmetics in the game.
There are 4 different possible tiers, legendary, epic, rare, and common. Not all items have legendary cosmetics available.
If I had to guess on the exact number, I'd imagine its in the thousands. I know someone who has opened about 1200 packs in all (excluding event packs). They have most in game content unlocked, but there is still a substantial amount that is not unlocked.