You don't have to have checking accounts at the same bank. Heck, you don't even need a checking account, but the person receiving the $$ needs some sort of an account.
For years, my wife and I have deposited checks or cash in the accounts of various relatives at various banks with no issue whatsoever. Currently, we put $500 a month via personal check into our son's account at US Bank....our account is at a local community bank.
We have a stack of the "counter deposit slips" at home. On the first day of the month, I write a check payable to our son, fill out the counter slip and take it to the bank. Endorsement is not required if it is being entirely deposited and there is no cashback.
Rarely, I am asked if I am the account holder. I always tell them no, it's my deadbeat son's account and they laugh.
The primary reason for asking is they will give you balance in the account if you are the account holder. They won't offer if they know you are not the account holder. But, most of the time, they just call me by my son's name ask me if I want the account balance -- I say no , I don't need bad news -- and they tell me to have a good afternoon.