Bluetooth pairing on Mac and Windows causes one to dismiss the other
The behavior you describe is indeed unusual, in fact, previous pairing should be lost regardless of the OS used. During pairing, Bluetooth host generates a link key which depends on the claiming device's MAC address and a random number. This key is the essence of pairing: two devices trust each other because they both know the link key. So, when you pair your device to the second computer, the link key gets overwritten and pairing to the first one is lost.

Since multi-pairing is not mentioned in BM77 datasheet, I assume it is simply not supported. You should take a look at different Bluetooth modules if you need it. For example, RN41/RN42 store up to 8 link keys in first in - first out fashion.