I have flown from NYC and LA to Ireland with dogs. Dogs that would have otherwise fit under the seat. Unfortunately they have to go as checked baggage or cargo. 

I have never quite figured out if this is some sort of international air transport law, or just airline policy. But, I was never able to locate a carrier that would take them in the main cabin.

 See Aer Lingus' policy on it here

https://www.pettravel.com/airline_pet_rules/aerlingus.cfm

If the process is still the same, it could be due to how they are picked up by a 3rd party in Dublin, and taken immediately to the vet facility to get checked out. You (or whoever is picking them up) will pick up your pet from there. 

I just flew over with a dog two weeks ago on Aer Lingus, and there were both cats and dogs on my flight. A cat should be easier since they will definitely fit under your seat. You need to notify whichever port you plan on flying into, the email is on the Department of Agriculture's site ( removed link ) 

This website walks you through everything you need to do, but basically your pet needs to be microchipped, receive a rabies vaccination post-microchipped and at least three months prior to arrival, needs a signed and certified EU health certificate no more than 10 days prior to arrival, and finally a tapeworm vaccination no more than five days prior to arrival.

Upon arrival, you will meet with an official at a time previously set up by the Dept. of Agriculture who will deem whether your papers meet compliance paperwork. 

Of course, make sure you notify the airline as well.

If you have further questions, I'm sure I can answer them. I navigated this whole process without much guidance and I would have loved to have been able to speak with someone who had been through the process recently.