The standard mains supply is ground referenced. If you are connected to ground somehow (eg. bare foot on a concrete floor) and you touch the live conductor, you form a path between live (which may be at ~110 V relative to ground) and ground, and will get shocked. If you connect the reference lead of a scope probe to this live conductor, you are shorting this conductor to ground, which may lead to damage to the DUT, the scope or you (see Dave's how not to blow up your scope video).
One way around this is an isolation transformer. An isolation transformer isolates the 'live' and 'neutral' wires from ground, so you can safely touch both ground and one of the outputs from the isolation transformer. Note that there is still a potentially lethal 110 V between the two outputs. You can also connect a grounded scope reference lead to one of the outputs, this will place this output at ground level, and the other at 110 V relative to ground. This means the outputs are no longer floating, and fireworks will ensue if you try to also ground the second output. The same applies to any transformerless DUT connected to the isolations transformer.
If the DUT contains a transformer and is double isolated (no ground connection), then the circuit will look like the top circuit in this schematic:
The motor will be floating regardless of the presence of the isolations transformer. The isolation transformer will only affect the primary side of the DUT power transformer.
A scope is usually grounded, that gives the second circuit. The output is grounded regardless of the presence of the isolation transformer. The same applies to any other grounded circuit. This can be bypassed by cutting the ground connection, but this is strongly recommended against for test equipment like scopes, and should also be done very carefully for DUTs with exposed grounded metal.