We have some customers who have two business accounts with the same name but one is Inc and one is LLC - different EIN's same signers. They are transferring funds between the two. Is this ok?

It is OK as long as you allow it to happen. I would want to know the purpose of the transfers however to unsure they made sense from a business purpose and did not generate a SAR filing. Maybe they buy and sell products and materials from each other??? Who knows without asking.

As to the fact they have both a corporation (Inc.) and LLC, there's nothing at all problematic with that. Lots of corporations form LLCs for business ventures they wish to keep separate from their other operations.

If I have an LLC and a corporation I set them up to protect myself from personal liability. However, I also set them up to contain risks; i.e. if the corporation gets sued, they cannot touch the LLC's assets even after the corporation's assets are exhausted.

However, if I merge the two entities' activities; e.g. transfer funds back and forth or open a common bank account, a claimant will "pierce the corporate veil" and say that the entities are really a single enterprise and that one is liable for the actions of the other.

It may not make a lot of difference to your bank, but if they are just moving money back and forth to wherever it's needed it's a pretty dumb thing for a customer to do.