We are thinking of taking a short flight from Dublin to Edinburgh. Does one have to go through the whole customs process to go between Ireland and the UK?
To my knowledge, the Common Travel Area is still in effect after Brexit, so in that case, No. It does apply really only for UK and Irish residents, so some report having to go through Passport control entering Ireland from the UK by plane if you are a US resident, but I do not ever recall having to present a passport entering the UK from Ireland.
However...things can always change.
Presuming you do not have a nationality that requires a visa to enter the UK you will see neither immigration not customs on a flight from Dublin to the UK. In reverse you would go through immigration though.