Replacing Doorbell Transformer
The simple answer to your questions.
Is it uncommon for the transformer to be mounted so far away from the door bell and thermostat.
The location is not specified in code and many contractors locate them at or above the chime unit or in an water heater / furnace closet. So it is not unusual.
Can a door bell and thermostat share a transformer?
Years ago it was not something I have seen pre 1980’s,
As we get into the y2k era I have seen furnaces and door bells sharing transformers more often and there is nothing in code stopping this.
I would do what @jsotola suggested and pull one side of the transformer red or white it won’t matter see if the doorbell still works if not and it will after reconnecting you can be 100% sure that is the correct transformer.
Get a larger transformer same voltage but more VA to run the ring I thought I have read 30va was enough so if you have a 20 get a 50 or with 2 doorbells converted to ring 80 or 100va as long as the voltage is correct it will not hurt to go bigger or as big as you may foresee using.