The meaning of "custodial" in crypto is covered in full in the entry Custodial vs. Non-Custodial Wallets, which compares the two models, their security and their regulation. This page gives a short definition and explains why the word is used on its own.
A custodial wallet, account or service is one in which a third party, such as an exchange, payment app or specialist custodian, holds the private keys that control users' crypto. Users see a balance and can ask for transfers, but the provider actually controls the coins, much as a bank controls the money behind an account balance. Custodial services can offer password recovery, customer support and regulatory protections, and are required for many institutions. The trade-off is counterparty risk: if the provider is hacked, freezes accounts or fails, users can lose access. A custodial wallet contrasts with a non-custodial wallet, where users hold their own keys.
"Custodial" is the adjective used across products, such as custodial exchanges, custodial staking and custodial stablecoin accounts, which is why the comparison article covers the term. The professional service is described under crypto custody.