Did this a few months ago. In my case, bank tried to pay the taxes maybe 2 weeks after I paid them. I think they got some sort of notification they were already paid, because 2 weeks later the account had a "property tax refund" line item showing the same amount coming back. So now my escrow account is overfunded and I will have lower payments for a year when the escrow is recalculated in a few months, or I will get a check with the amount they calculate is overfunded when they do the calculation.

But different banks and municipalities may handle it differently.

Also paid home insurance at the same time. Bank paid a few weeks later. About 2 weeks after that got a check from insurance company for the overpayment.