Are pets/mounts roster-wide?

[Kargor]

The thing is, I noticed those "Twitch drops" not too long ago, and got 2 pets and a mount from it. So far, so good.

Today, I'm logging in... and I get three "product inventory" items I can claim... 2 pets and a mount. The pets come in different colors anyway, so if I claim them... I could get another color for each. The mount is useless, though, since it only comes in one color IIRC.

Or is this meant for future characters?

[Saphirro]

AFAIK they should be

[DeiGratia]

Yes when you make alts just pick a different color and you can collect them all like Pokémon