In most cases it isn't a literal subsidy. Fannie Mae acts as a government-sponsored market maker to purchase home mortgages from lenders, then package and sell them to investors. By providing an "artificial" source of liquidity and taking on significant balance sheet risk they keep mortgage interest rates much lower than they would be in a free market system.
Mortgages are subsidized by the federal government. Housing supply is limited by state and local governments. Those are separate entities which often work in conflict with each other.
Doesn't matter who is limiting the supply though? Subsidizing something with artifically limited supply will automatically raise the prices and hurt ppl who they are intending to help. What am i missing here?
https://www.fanniemae.com/about-us