The schematic in my Haynes manual suggests that the aux fan should work only with ignition ON. However, maybe whoever wired your system changed it to work even after you have turned off the car. This can be a good thing and I know that I've run across cars in a parking lot with the engine off but the fan still running. I suppose as long as it doesn't stick ON and drain the battery it is OK.

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"Buster" in the '95

Our all-Diesel family
1996 E300D (W210) . .338,000 miles Wife's car
2005 E320 CDI . . 113,000 miles My car
Santa Rosa population 176,762 (2022)
Total. . . . . . . . . . . . 627,762
"Oh lord won't you buy me a Mercedes Benz."
-- Janis Joplin, October 1, 1970