My boxes 15 - 17 on the W2 look like this :
TurboTax is asking for an employer ID number for the Ohio line. Is it the same for both states? Does this mean they paid Kentucky instead of Ohio, where I live? My Indiana one is the same.
You can omit the Ohio entries on the W2 - you will have to file a nonresident tax return for Kentucky and Indiana.
You may want to review What is a state reciprocal agreement? - TurboTax Support - Intuit and discuss with your employer.
Q. Does this mean they paid Kentucky instead of Ohio, where I live? My Indiana one is the same.
A. Yes.
Q. TurboTax is asking for an employer ID number for the Ohio line. Is it the same for both states?
A. No, but try using it anyway (or the federal ID#). It's doubtful that your employer has an OH ID (otherwise he'd done it right).
Ohio has a reciprocal agreement, on wages, with all neighboring states. No state taxes are withheld or due and you do not normally need to file a return for the other state (MI, PA, KY, WV, IN). But OH will tax you on your other state income.
If other state taxes are mistakenly withheld, you then have to file a return, for the other state, to get a refund. OH will not give you a credit. You should ask your employer to stop withholding so you don't have to file a (KY, IN). return every year.
On the IN non-resident return, be sure to indicate zero Indiana income.
KY has a specific form (740 NP-R ) for a reciprocal state refund. Rather than paying TurboTax for KY software, you should try filing that form by hand. It looks pretty simple.
https://revenue.ky.gov/Forms/740-NP-R.020521.pdf
However the reciprocity agreement does not apply to local city or county taxes. You can not get those refunded, but if you live in an Ohio city with a tax, they may give you a credit or partial credit. Many do. You need to check with your city.