The short answer is: yes, you may do that.
BroBizz is not associated with a particular car or number plate. It is an independent means of payment that is linked to your household.
Therefore, you can safely bring your bizz if you borrow the neighbor's car when your own is in the workshop.
However, special rules apply if you are going to Norway. In Norway, it is important that the car's license plate is registered and updated on the BroBizz you have with you.