I messaged United about it but I'm not holding my breath for a prompt response.
Here's what I got from Aer Lingus:
Passengers holding single tickets which include both an Aer Lingus flight to/from North America and an Aer Lingus flight within Europe will be subject to transatlantic baggage allowances for the entirety of their journey. Bags will be automatically checked to their final destination.
Passengers holding seperate tickets on Aer Lingus services will be subject to the individual baggage allowances of each flight. Aer Lingus will thru-check, as a courtesy checked luggage onto a seperately ticketed Aer Lingus flight upon presentation of the onward reservation to the check-in agent.
I asked for further clarification since even though both flights have Aer Lingus flight numbers, neither are actually Aer Lingus flights.