That's exactly how Tor got created -- it was originally funded by the Navy. The DoD version of Tor _is_ Tor.
Also, it's entirely possible that the government is using the same software (or a slight fork) on a different network, but using the public deployment of Tor as a way to easily get public review of the cryptosystem.
Furthermore, if only the DoD/whatever used Tor, "someone with Tor traffic going from/to them" would be a good-enough metric to locate DoD emplyees / government spies / etc. The idea is, that's why Tor continued to be developed not as part of a Naval Research lab project, but rather as a civilian project.
Also, it's entirely possible that the government is using the same software (or a slight fork) on a different network, but using the public deployment of Tor as a way to easily get public review of the cryptosystem.