A transformer usually has 2 coils - a primary & a secondary. There is no physical connection between the 2 coils. The primary induces a voltage in the secondary winding, but does not physically touch it. Thus the secondary side is isolated from the primary side. But the isolation is imperfect due to capacitance between the coils, etc.
An true isolation transformer adds shielding between the coils for more complete isolation between the 2 sides. The shielding has its own connection to earth ground.
A variac is an "auto" transformer -- "auto" meaning "self" or "one". That is, it has only one coil. Hence it is confusing for it to be called a transformer. It is more like a variable inductor, with only one coil and a variable center tap. Because there is no secondary winding, it provides no isolation for the DUT (Device, or amp, Under Test) from the AC wall power supply.
From what I can gather, isolation of the DUT is, or was, important for at least 2 reasons. Old transformer-less amps might have a live chassis, which is dangerous to work on without isolation. Also, some old test equipment like VTVM's and 'scopes, if they have a 3-prong power plug, will create a short circuit when their ground test lead is connected to the chassis of the DUT. Isolation prevents this short circuit if the ground test lead is used. Modern test equipment avoids this problem by having an internal floating ground. Floating grounds are dangerous to the user; so the test equipment is housed in a plastic, not metal, case. This protects the user from a ground fault in the modern test equipment.