Please check the video I provided above that is how I withdraw funds out from the vault.
That's coinbase vault, not coinbase multisig vault.
Coinbase vault (which is used by you) is a custodial service and coinbase has full control over the fund. There is no real co-signer in coinbase vault. When you approve the withdrawal through your email address, you only tell coinbase that the withdrawal request has been made by the real owner and it's not that you are the co-singer.
Coinbase multi-sig vault (which is no longer supported) was a 2 of 3 multi-signature wallet in which the user had access to two of the keys and had full control over the fund.
From Coinbase help:
Note that the multisig vault is different from the regular Coinbase vault, where we handle security on your behalf so you don't have to.