From what I understand USCIS is funded entirely by fees. The FAA has the Aviation Trust Fund, and last year a big legislative push was to make it so the ATF can fund us during a shutdown. But as it stands, the FAA is funded by normal appropriated money from Congress, and as a side detail sends enough money to the general fund from the ATF that it comes out to nearly if not entirely cost-neutral.
Which is to say, our situation is not like their situation, despite the fact that the ATF is in a sorry state due to reduced fuel and passenger fees.