Regarding banks, here’s what the OCC says:
May a bank take money from my deposit account to make a payment on a loan that I owe to the bank?
Usually, yes. Generally, a bank may take money from your deposit account to make a payment on a separate debt that you owe to the bank, such as a car loan, if you are not paying that loan on time. This is called the right of offset.
In some situations, the bank can exercise the right of offset without letting a customer know in advance that it is going to do it.
However, federal law limits what a bank can do in some cases. For example, federal law won’t allow a bank to offset your deposit account to pay off your consumer credit card account.
This is different from cross-collateralization where, for example, if you have a secured auto loan and an unsecured personal loan with the same FI and subject to a cross-collateralization agreement, the FI can repossess the vehicle securing the auto loan if you default on the unsecured loan (even if you don’t default on the auto loan). And as you said, this kind of clause appears to be universal or nearly universal with credit unions, but it doesn’t appear anything stops a bank from including it in their agreements (it’s just uncommon).