Isn't there gonna be a very high amount of output inductance in series with the load? Is this ever a cause for problem? Just curious because I didn't seem much information online about it.
You don't see much information because it's not a problem.
A Variac is just an adjustable power transformer lacking isolation. It has all the problems of, say, a 120 VAC to 24 volt transformer, which is just about nothing, or at least nothing unusual.
The main inductance of any transformer (not just a Variac) is not in series with the load, it's just the magnetizing inductance, which determines the magnetizing current.
There is a small leakage inductance in series, but that's usually negligible for normal power transformers.
There is some loss in the wiper because it always spans at least two of the windings. The brush is carbon so that the short between the windings is not zero ohms.
I would expect that a good data sheet or an application note for a "Variac" will say something about the losses.