All inductive transformers are isolated except "auto-transformers" which share the conduction winding.
However all transformers also have a coupling capacitance which can lead to a sense of false security if you touch an isolated secondary and expect no current to flow.
This is because two capacitors can be a step down transformer as well, where the smallest capacitor takes the biggest step, and compared to a grid transformer, you are the smallest capacitance.
The bigger the transformer, and its dielectric constant in the insulation, generally, the bigger the coupling capacitance which tends to be much bigger than the average human capacitance from air to your finger tip of 100pF.
Hence all distribution transformers (DT) are earth grounded on the neutral of a Y or split phase output.
The isolation transformer helps protect measurement equipment from catastrophic ground clip connections. Ferroresonant types maintain a constant voltage but are far worse for conducting line transient noise.
Isolation transformer are perfect solutions when needed for equipment or human safety. But noise can be worse. Worse yet, the insulation can charge up to lethal levels without any resistance to ground.
The down side is potential greater common-mode noise if the unit contains a lot of switched currents. This is common in laptop chargers because the DC output is floating and external mic's often get hum from the charger until the person touches ground with a finger or the laptop is grounded to an external monitor.