ISTEA - Intermodal Surface Transportation Efficiency Act of 1991
It is possible that MUPs in the Maryland, Virginia, and DC areas built before 1991 did not use federal funds. At that time, as well, the (then known as) Bike Paths were under the respective Parks departments.
The 1990 Earth Day - Bike to Work day organized an event with congressmen and senators pushing the passing of ISTEA (which I attended). Each of the Maryland, Virginia, and DC bicycle advocacy groups pushed for the funding for bike paths to increase bicycle transportation. ISTEA passed, and the groups began to push state legislators to get the federal funding by providing the states share of the funding.
I believe any new paths/MUPs built after 1991 did use federal funding, even if the real intent was for the paths to be recreational. This is common throughout the country - states simply lied about the intent of the paths. The operational authority simply remained with the departments (parks) that had the responsibility prior to ISTEA.