You name your token when you mint it, but the unique part lies in the policy id which is a hash of the policy script. If you look at the testnet now there is a lot of tokens with the same name but under different policies, so there is no way to claim a certain name right now. It might be possible in the future in the same way that you can “claim” your pool ticker if someone copies yours, though it is not actually removed from the chain, only not listed at some of the third party softwares.
Hope it answers some of your questions!