Can a hot tub be plugged into a 240v 3 prong receptacle?

If your tub only needs a 30 amp 240 volt 3 wire service ,then yes. Most tubs needs 4 wire, with a neutral, and most need 50 amp service.

240 v hot tubs usually do not come with a plug, did you install this plug or was it factory?

It also depends on how that receptacle was fed (insulated ground?), and whether your tub is 120/240v which would require a neutral (4-wires).