Yes, the most common types of variable voltage transformer use the same winding for both the mains input and the output, so the isolation between input and output is essentially none, as one end of the winding is a common point. more complex types use either a second transformer to provide isolation between mains input and output, and there are some designs that use a second winding wound around the core, isolated from the variable winding with some robust insulation, so that electrically the windings are separate. Both options are not commonly encountered, and are quite rare.
I however have used a variable voltage transformer with a separate transformer to provide both mains voltage reduction and isolation, using a 50VA toroidal transformer to provide 12VAC to the variable voltage unit, so I could get a variable 0-12VAC supply at up to 5A, as the variable transformer I was using, a unit otherwise unusable as it was originally designed for 115VAC 400HZ avionics use, was not meant to operate on 50Hz without serious derating of the input voltage, and 12VAC was well within the VHZ product, so the core would not saturate. Still use it as a benchtop adjustable AC supply, though for a number of years it did sterling service driving a 12V halogen lamp, as a dimmer to tame the light output as needed. No Rf or switching noise from it at all, as the control was totally linear with no semiconductors involved.
