Make it a collar or anklet next time

If male: chastity cage.

Does Teleportation have somatic components? Can they make the intricate gestures with a dislocated thumb?

If we're talking Teleport the 7th level (in 5e) spell, then nope, no somatic components. Teleportation spells in general about 50/50 whether it does or doesn't require somatic components. Fewer still require material components, which I expected to be more of an issue, but apparently not.

Do all D&D characters have thumbs?

Obviously this one does, but it makes me think that a thumb isn't a requirement to cast all spells.

I just checked the list of playable races, and yeah they do. Even the plasmoids and the bird ones have hands with thumbs.

I could see dipping warlock sorcerer for subtle spell, negate the somatic components. You could even be like that one guy from Doctor Strange, no hands at all.

Opening a tavern after retiring from adventuring is so out of fashion.

I'd open a prank/novelty/gag store and sell little wooden thumbs to the unthumbed.

I'd also sell middle fingers but only to bards.