Edilocal
Thru check is not (NOT) related to travelling on the same airline throughout. It isn't even related to the airlines being part of the same alliance.
If you are making a connection between 2 airlines the requirement is the existence of an "interline agreement". But not all interline agreements are created equal. There are ticket agreements, baggage agreements and checkin agreements. You can have one without the other and they can be "one way" e.g. airline A will accept a ticket issued by airline B but airline B won't accept a ticket issued by airline A.
Thru checkin assumes a connection between airline computer systems because if the airline you are checking in with is going to give you a boarding pass for a different airline they need to get the seat number from that other airline so thru checkin has a cost attached and that means the amount of traffic has to be worthwhile. Significant oversimplifiaction there!