In 1988, the ticket may have been printed on cardstock or handwritten on a carbon ticket set. You could purchase a ticket for another person and have it mailed from the ticketing agency to the passenger, or they could show up in person and pick up the ticket. The ticket was like cash, so it wouldn't just be sent regular mail, but some secured method. Sometimes a reservation would be made, funds sent to the passenger via Western Union or bank transfer and the passenger would then pay for and pick up the ticket at the airport.
Not only would a cash ticket not set off alarm bells, it is quite possible that they would pay with a check (at least in the U.S.).
The ticket must have a name on it. It was exceedingly rare for ID to be checked on a domestic flight - I know someone who traveled on his mother's ticket as a 19 year old male. The gate agent simply glanced at the ticket and said, "You don't look like a Catherine." It was also very common to book tickets as "C. Smith," as that could be Catherine or Charles.
At that time, there were no (or very few) restrictions on underage passengers traveling alone. You *could* pay for an "unaccompanied minor" and have the child handed over at the final destination to another party. Most people just put their kids on the plane and told them not to talk to strangers -- even internationally.
As for travel between France and Ireland at that time - would you need a passport? Borders around the world were quite open, so it wouldn't surprise me if they just required simple ID. (At that time, US citizens could travel between the US, Mexico, Canada and the Caribbean with a driver's license. It was smart to travel with a birth certificate for a child, but not always a deal breaker.)
Baggage restrictions were also much looser. If you could cram it under the seat, or in an overhead, you could take it on the plane. (Excepting, of course, guns and knives with a blade over 4") My husband took a spearfishing gun as a carry-on in the late 80s without any issue. (No, it probably didn't meet the requirements at the time, but it was in the bag and no one cared.)
People didn't dress as nicely as they did in the early days of air travel, but you weren't likely to see adults in their pajamas. (Wow, that makes me sound really old...)
My family owned a travel agency for many years and I was blessed enough to do a lot of travel at that time. It really was a different experience than it is today.
Best of luck--