Nope ... this was a taxable fringe benefit on which fed/state/FICA taxes should have been withheld already.  By virtue of them giving you money to cover the move is a good thing (and was not required of them)  so hopefully covered the entire cost.  Would you rather pay the taxes on the cost of the move or pay for it completely out of your pocket ?  If you think you were short changed then this is a conversation to have with your boss. 

Q. Should not my employer reimburse me for the taxes I paid on the second W-2?

A. No.

Moving expenses are no longer deductible.  So, any reimbursement you get from your employer is additional taxable income to you.  Recognizing this, many employers also provide a Relocation Income Tax Allowance, on top of the reimbursement for actual moving expenses. But that allowance is also taxable income to you.  

If your employer miscalculated the amount of tax, then you may see  a refund when you file you tax return. 

The employer could gross up your bonus if they wanted to, but that is up to them.  

After tax reform, relocation expenses are not tax deductible to the employer or employee.  It is treated as if you received a cash bonus and is taxable as such, no matter what you used it for.  

If your expenses were, for example $5000, the employer could pay you $8000 to cover the taxes, but that would be up to them.  What did you agree to in your contract?