A vanity IBAN is an account number in the international IBAN format that has been selected, in part, to be easy to remember or to spell something. A business might ask for an account number ending in a run of zeros, or, in countries whose format allows letters, one that contains its name.
Only part of an IBAN can be personalized. The country code and the bank identifier are fixed, and the two check digits are calculated from the rest, so they change with whatever is chosen. The freedom lies in the account-number section of the basic bank account number, and only if the bank or payment institution is willing to assign it on request and the number is unused. Providers that issue virtual IBANs from their own number ranges are best placed to offer this. A vanity IBAN works like any other and still passes an IBAN checker. The crypto equivalent is the vanity address.