Help within FCO

This whole plan, to me, is an absolute recipe for disaster. 1h15min is absolutely not enough time to deplane, go through passport control, and get to a different airline on a totally different booking.

What happens if you miss your Aegean connection?

Unless your US flight arrives quite a bit early, Aegean Air check-in will probably have closed before you can get to the counter. I'm not certain whether using the Transfers lines would allow you to short-cut the process (hopefully someone who knows for sure will post), but that might be your only hope of making the connecting flight. If you need boarding passes to use the Transfers lines, then you'd need to check-in online either before departing the US or immediately upon landing in Rome. Without changeable tickets that would allow you to take a later flight, checking luggage could wind up being the least of your worries. I'm not trying to be mean, just realistic. Good luck!

My sister has a layover of 1h 15min from Houston TX USA to Frankfurt Germany but booked with same itinerary and same airline. Why would airlines suggest those flight options if it is not possible?

The key difference here is one ticket vs separate tickets.

When a connection is purchased as one ticket the airline(s) involved are obligated to Germany’s you to your destination. If you miss your connection then they will get you on the next flight with available seats.

When you purchase separate tickets there is no such requirement. So if you’re delayed getting to FCO and miss your Aegean flight, Aegean Just sees it as if you didn’t make the flight. So unless you bought a ticket that allows changes you need to find a new way to get to your final destination.

When you buy a single ticket for 2 flights with the same airline or code-share airlines, you're checked in for both flights at your first airport and your luggage can be checked to your final destination. The airline is obligated to get you and your luggage to your final destination whether on the booked flight or another flight. The time between flights is such that most people changing flights will make the second flight if the first flight is on time, but the airline isn't worried because it can simply put them on a later flight if they miss the first one. The second flight also knows when such connecting passengers are running late and can delay their departure to accommodate those passengers if circumstances allow.

When you buy 2 separate tickets on 2 separate airlines, there's no communication between the airlines. Even if you've checked in online for your second flight, that airline doesn't know whether you're running late or just decided to not show up. They'll depart on time if all other passengers are there. It's "possible" you can make the connection, but you assume all responsibility to be there on time and to make other arrangements if you miss it.