Baggage policy Etihad or American Airline?

As your final destination is in the US, the US DOT Regulation 14 CFR 399.87 applies:

For passengers whose ultimate ticketed origin or destination is a U.S. point, U.S. and foreign carriers must apply the baggage allowances and fees that apply at the beginning of a passenger's itinerary throughout his or her entire itinerary. In the case of code-share flights that form part of an itinerary whose ultimate ticketed origin or destination is a U.S. point, U.S. and foreign carriers must apply the baggage allowances and fees of the marketing carrier throughout the itinerary to the extent that they differ from those of any operating carrier.

So if your itinerary does not contain any codeshare flights, it will be the baggage allowance for the first leg that is relevant. In your case, Etihad's rules for the trip from Lahore to Rome are the ones to apply. (On the return trip, it would be American's allowance for the journey from Manhattan to Rome.)

If your itinerary contains codeshare flights (e. g. if the American flights are showing with EY flight numbers), then the marketing carrier's (here Etihad) allowance applies throughout. This is important as Etihad's allowance from Pakistan to the US is more generous than their allowance from Pakistan to Europe.

However, I think all of this is only valid if your baggage is checked through to MHK. If, for example, you have to pick up and re-check your baggage in Rome, then American might apply their own policies for the remaining two flights.

For the sake of completeness: If the origin or destination is not in the US or Canada, then the baggage allowance of the most significant carrier according to IATA Resolution 302 applies, but this is not important for you.