Actually the entire Class 172 fleets all share the same BSI coupler design so to be pedantic there's no reason why a Chiltern Class 172 can't work in multi with a London Overground Class 172 or a West Midlands Trains Class 172 with a London Overground Class 172 apart from the fact that obviously you can't walk between a Chiltern/London Overground Class 172 which don't have end gangways and a West Midlands Trains Class 172 which does have them.
That's from the relevant ROSCOs too
This shouldn't be a issue because the UK has operated Class 150s with Class 14Xs before in multi.
Class 168s have slightly different BSIs to Class 170s I believe, maybe the pins are in a different place?
Exactly - you can't couple 16x to 14x, 15x or 17x because of some either very childish or very shrewd (delete as appropriate) decisions made back when the Turbos were first introduced to make sure they were kept in NSE territory. Although the basic coupler design is the same, their are differences with the pins making them incompatible.
The original WMT 172 fleet was designed to be compatible with the rest of their fleet, so the couplers are set up for 15x and 17x. The Chiltern 172 fleet needed to be compatible with their existing fleets of 16x, so their couplers are slightly different. My understanding, although I'm happy to be proven wrong, is that London Overground also went for the 16x pin arrangement for their 172s, so they will need modifying if you want them to work with any thing else at WMT.