asdm is simply a gui interface to your console configs and nothing more.
asdm goes to flash memory / disk memory and it is a .bin file in the directory.
you cna use it for both routed and transparent mode and also for multi context / virtual domains/ virtual firewalls.
asdm will stay the same for all contexts, but I believe if you create different usernames under each context and try to login to asdm , it should launch only asdm as per user rights only for that context.
