It's not a usage I've met, but I think I know what's going on here.
There's a meaning of 'check in' that can connote 'go through intial formalities': 'When delegates arrive, they should check in at Reception."
An inspector of some kind - say, quality control - may 'check out' or 'check over' something.
In the case you quote, it seems that all these meanings are being rolled together. There is a powerful matriarch to whom every young lady must be presented.
Informal arrangements like these were formalized at one (not-too-distant) time in the 'Coming out balls' attended by Débutantes - Wikipedia, the free encyclopedia .