Generally, the taxes withheld in a situation like yours would typically be to withhold the taxes from PA, not NY. However, NY is from the few states that fall under the category of "convenience of employer rules" and as such taxes would have been withheld from NY rather than PA. As such is the case in your situation, yes, NY taxes should have been taken out. When you prepare your NY nonresident IT203 at the end of the year, you will deduct your NY taxes withheld for the remaining 7 months from that IT 203 tax liability . Whatever tax liability you will have leftover on those NY taxes will consequently go as a credit to your PA return. Obviously, the 5 months tax that was withheld in PA which is your resident return with appear as a deduction from the amount you owe on that PA return as well.
I guess if an underpayment penalty gets calculated at the end of the year due to underwithholding for those 5 months, you will bring it to your employer's attention to reimburse you for that.
I wish you all the best.
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