Why would upgrading from Lion to Mountain Lion more than double the size of my Time Machine backup?

Probably simplest to just restart your TM backup.

See Pondini's Time Machine FAQ for dealing with such things: http://pondini.org

If it is working as expected, what do you need with the old backup?

If you want to be able to go back to where you were prior to an upgrade, isolate that backup and don't use it again. As far as Time Machine can tell, the whole drive changed.

The link I posted contains probably everything that anybody knows about Time Machine and how to manipulate it to work as you would like as opposed to how it was designed to work. There are FAQs to continue a backup as opposed to starting fresh, but I wouldn't recommend it for the reasons above. Pondini probably doesn't, either.

If you are going to do a wholesale upgrade of the OS, I would create a bootable clone using something like Carbon Copy Cloner or Super Duper. A bootable clone on an external drive can have you back up and running instantly if the upgrade fails in any way. If the upgrade goes fine, store it away in case you find the current system isn't to your liking.

If you use a different computer name with each OS X installation, Time Machine will dutifully start a full backup of the new computer name to an adjacent Time Capsule location. If you had other mounted (internal/external) drives that were not explicitly excluded in Time Machine preferences, they got backed up too with the new Mountain Lion install.