Do Air Traffic Controllers have to remember stall speeds for different aircraft?
No because aircraft are categorized by their speed at the runway threshold (1.3 times stall speed).


VAT —Speed at threshold used by ICAO (1.3 times stall speed in the landing configuration at maximum certificated landing mass)

By knowing the category, ATC is able to use appropriate speeds. The category is not actually listed anywhere, so the controller will have to know the approach category by heart for each aircraft type. E.g. if the controller sees "B738" as the aircraft type in the flightplan, they have to know that the corresponding approach category is C.

More:


ICAO Doc 8168 PANS-OPS Vol 1
14 CFR Part 97.3 (for the US)