This behaviour is mostly limited to credit unions.
FDIC rules (banks) prohibit offset of deposits to pay credit card accounts (except for explicit arrangements such as a savings account specifically tied to a secured credit card). Credit Unions are not limited by this rule -- therefore credit unions see an active checking account with them and probably receiving your direct deposit as a buffer. They could cover some losses by freezing funds and applying them towards your card debts. A large bank account balance with say Citi is no different than a large balance with any another bank. It might allow them easier verification that you have those assets, but it gives them no special rights to it. A large balance in your Alliant CU account gives Alliant some rather nice special rights to those funds in the event you stop paying your card.
For tax and rule of law purposes, the major banks also have their credit card operations segregated into a loosely related affiliate of the retail banking operations. The person making underwriting decisions on your credit card application is unlikely to have access to anything more detailed than some flag that you are a retail banking customer. Getting one side to talk to the other is usually a big obstacle. I've had instances where I've had to conference a supervisor from each side together from my end, because it was the only way to get them to talk to each other.
Credit Unions are much smaller and with a few clicks they can, and probably do, look at all the business you do with them. Most frequently you see people mentioning a history of direct deposit in their CU account being used as income verification for their loan application and things like that.
