Isn't that surprising? (Please, don't say "No".)
Two observations on countability:
1. I've found numerous examples of "to be under obligation to do something" and not a single one of "to have obligation to do something".
2. I've found no examples of "to have obligation/duty to do something" and several ones of "to have responsibility to do something".
I can't explain this difference.

Retired English Teacher