[floomk]

Why do these companies have the right to detain someone?

[ceejayoz]

They don't. You're free to leave the airport, forfeit your flight, and be banned from the airline.

They are, of course, not going to go out of your way to remind you of that right when they say "please come with us".

[awb]

They don’t.

If a non-law officer says “come with me”, you don’t have to, but people do it anyway.