We continue to struggle with customer requests to access their Business and Personal Accounts together through online banking. We have made it policy through the years to keep these separate, but have allowed some on an exception basis. We also get numerous requests to pull multiple Business Accounts under different names/Tax ID Numbers together online for a common owner access.
This becomes a real challenge when someone has worked hard to get someone to move their accounts to our bank, and the customer insists that they were able to combine online access at their previous bank.
Are there situations where allowing this combined access would be acceptable? Are there steps that can be used to mitigate any potential risk?

You should discuss this with your legal counsel as to what mitigation techniques are available in case things go terribly wrong.

When you allow combined access, are you subjecting the accounts to the consumer rules, or the business rules? How do you verify that accounts with different names and tax ID numbers have a common owner?

It certainly is not clean or controlled and subjects all the funds to loss in the event of an account takeover or loss of the credentials to a scammer. It can tend to allow greater access than needed when the owner wants "A" to have access to some, but not all of the accounts, but loans A his password anyway.

Instead, I'd recommend they maintain tighter controls and/or invest in a password management program that allows the owner to give each user access that way. They logon to the PW mgmt program and it logs them into online banking or whatever based on what is entered into that program. It saves keyboard entry and memorization. Interbank transfers need to be limited between the business and personal account because the bank doesn't want to hear the business was sold and the account needs to be transferred to the new owner, and a few months later, "oh, he was getting money from my personal accounts because the bank forgot to remove that access..."

I hate "well my other bank..." as that was so common when someone brings in a new MMDA depositor who wanted unlimited check writing, "because my last bank allowed it."