This other federal money, sometimes referred to as the "federal payment", comes from Congress as part of their DC budget appropriation process, part of the federal appropriation/budget process. The DC appropriation is approved along with the rest of the annual budget and is signed by the President. Once approved, the funds are included in the city's accounting system and show up in the budget as Federal funds. The DC government must file special reports for these monies.
Congress also appropriates funds to nonprofits; this money flows through the Office of the Chief Financial Officer. Yes, these are earmarks. Special reports are required for these funds as well.
So why do we care? Well, this is just another bit of knowledge that explains how the city is funded and why we get some of the money we do.