Yes, it is possible, and great fun!
One way to synchronize apps is using Korg's WiST technology. This allows apps on multiple iDevices to synchronize over WiFi. They maintain a list of compatible apps on their site.
Another method is to slave all of your iDevices to a single MIDI clock (e.g. a drum machine that sends clock over MIDI). There are multiple ways to get MIDI into your device including the Alesis IO Dock, the iRig MIDI, and the Line6 MIDI Mobilizer. This is a little more hassle than the WiST approach because you have to wire up a bunch of cables, but it opens up a larger group of apps since there are many apps that support CoreMIDI that do not support WiST. When shopping for apps in the App Store, look for any that say they support CoreMIDI. This page has a good starting list, but is by no means comprehensive.
Finally, in addition to this multiple-iPad fun, you can also make use of MIDI between apps on a single iPad. e.g. Running a sequencer that is controlling a synth app running in the background. Not all CoreMIDI apps support it, but many do. I've been having a great time using the Genome sequencer app to control synths such as Animoog.
It's a great time to be a musician and have an iDevice....much is possible now that was impossible only months ago!
Why not tether the iPad to iCloud, then it doesn't need a computer at all? If you do that and turn on backups, you can then set it to manage music (or media, if you do videos too) manually, and then you can drag stuff on and off from either computer, and none of your purchases will be lost.
When you plug into the new computer it might ask you to authorise your account but you can have up to five devices or something so that shouldn't be a problem.
Best Answer
No, the two installations are independent of each other.