Are power banks allowed by some airlines but not others?

Aer Lingus has no process to check passengers baggage. The airport does that and - presumably - applies the same rules to all passengers travelling on all airlines.
 
To be clearer, as long as they're not excessively powerful they're allowed in cabin baggage, but not checked baggage. I'm pretty sure I actually bought one in a shop in Dublin Airport after passing through security.
 
Slightly OT, but was on a Jet2 flight recently and all seats had USB power points available. Very nice touch indeed.