It is a 3-phase BLDC motor. Similar types can be called synchronous motors, or permanent magnet AC motors. Exact winding shape defines whether it's called "BLDC" or "PMSM" but this is just details and doesn't matter much here.
You should measure the currents of at least two phases simultaneously to get the complete image of all phase currents.
What you show works but you need to deal with either low-value shunt, and low voltage over it, likely swamped by noise; or relatively large-value shunt, and a lot of power dissipation, and intrusive measurement (ESC may work differently because of the resistor).
I'd recommend a specific current sense amplifier coupled right next to the shunt with tight layout, and scope that output signal.
A proper ESC itself has such current measurement, so it does not produce too much current. Obviously, this is some shit ESC and a current measurement alone won't give you much information.
There are basically the following typical causes of failure:
1) Overcurrent combined with the lack of current sensing/control,
2) Overvoltage due to poor rating of parts, or poor layout (unsnubbed switch node ringing), exceeding Vds rating
3) Failure to drive the gate properly under all conditions, causing operation in linear range and quick, excessive heating
All of the above can happen at microsecond timescale;
4) Lack of proper thermal design; longer time-scale (tens of seconds to minutes) overheating and finally destruction.
You can kinda estimate the cause #1 with the measurement you propose.
Generally, in a system which provides a lot of inductance (the motor windings!), current changes slowly (no "spikes"), but voltage may change radically and quickly (voltage spikes). This is why controlling the current in a motor drive is almost trivially easy (a few dozen kHz of control bandwidth does fine, just measure and feedback, even a badly tuned PI loop often does fine), and which makes failing to do that properly very awkward.