That Ryanair don't have anyone on the Aer Lingus board is irrelevant. Large capital purchases have to be approved by the shareholders. So, let's say Aer Lingus wanted to buy 15 Embraer 195s so offer more frequent service to UK airports than can work in an A320, Ryanair could block the purchase. It's not like they control Aer Lingus, but they can significantly influence Aer Lingus's business strategy.
Which has absolutely nothing to do with UK Government as both are Irish companies.
UK Govt would tell French to mind its own business if 2 UK companies had stakes in each other and France wished to declare what the companies should do.
The clowns in UK Govt don't realise they are now setting a precedent that allows other Govts to interfere in UK companies.
As for large capital purchase needing to be approved by shareholders ! Fraid not always the case as that is regarded as day to day management which they can do without shareholder approval.
Perhaps UK Govt should start looking at Easyjet where a large shareholder is directly interfering publicly in the day to day management of the airline including forcing directors out.