In CLI definition of bridge has also option called "auto-mac" and if set to "no" (together with static setting of "admin-mac") MAC doesn't change over reboots. WebFig doesn't show "auto-mac" option explicitly, might be set implicitly when one unfolds the "Admin. MAC Address" window and sets the MAC there. I'm not sure about winbox though.
By default, bridge assumes MAC address of the first member port when it starts to assemble. Interface initialization delay might scramble the order, configuration changes (e.g. removal of an interface from bridge) might do even more serious damage (broken connection due to bridge changing its MAC address).
When setting bridge MAC address by hand, you should not blindly copy current MAC due to the possibility of configuration changes mentioned above. But you can assign any locally administered MAC address
, just be careful it is unique inside your ethernet network. A good guess is to take current MAC address and change second most significant octet (second digit of the MAC address). Just verify there are not virtual APs defined on same unit that might select the same MAC address.