Imo i'd say 'no'. I've always heard it referred to as a tender. Even when some were saved as water cars for fire fighting, they've been referred to as 'water car that was originally a tender from a steam locomotive'.
I'll second the "no". A tender is a component of a steam locomotive (when equipped, tank engines need not apply), not a separate railcar.
Is it just a canteen, or does it carry fuel oil as well?
Another unpopular opinion from across the seas. In Russian terminology it is indeed a car. I think I heard somewhere a definition вагон-тендер which translates literally as tender car, however official Soviet/Russian encyclopedia of rail transport from the 80s I own describes it as: "a special carriage (platform, car) attached to the steam engine which carries reserves of fuel, water, lubricants and instrument". So apparently here it is viewed as an attachment to the engine rather than part of it. I know, sounds kinda counter-intuitive after I spend a few days trying to explain why in domestic practice a locomotive section is not a separate locomotive, despite it looks like it, but a modular part of a bigger whole.
That actually makes sense.
I think that, for at least Canada, the tender doesn't have a separate road number so it is seen as one part of the locomotive proper. I know that in the earliest days of the CPR they put the locomotive number on the tender as well.
How about this, if it uses a standard coupler it's a car, if it uses a semi permanent drawbar it's part of the locomotive package.