Can Mom gift dollars to son to care for her?

Are they saying it's a gift ? 

Does his POA permit making gifts?

"My question is, can this customer gift himself and his spouse 17k from Mom, or is it better to claim these dollars as income?"

With a full powers POA, he can just take it all. But, the gift limit (to each) is not going to cover "take 1k/week" and they are losing out on Social Security credits, if that matters. So, what about that shortfall?

"They want me to do Mom's taxes"

You could look into Mom having hired "household help."

"She moved into their home full-time last May, so Mom's assets have paid the customer approximately 28k for 2023."

And is she a dependent for them, or covering her own costs? She would contribute to the general costs to run the household (utilities, food, etc) and her lodging and personal effects (clothes, etc) as well as any personal direct costs (medical, etc). Are they doing Both? Charging her rent, room and board, personal maintenance, and also "wages?"

What's missing from this discussion is whether there are any siblings and, if so, how they feel about it.  

Withdrawals from the Bank of Mom are clearly not gifts, but the issue for you is whether they are some form of taxable compensation.  Some of it appears to be sharing of household expenses.  The rest of it falls into a gray area of family transactions that are overlooked most of the time.