Is the BLDC motor and "brushless AC servomotor" same thing?
BLDCs are a type of PM (permanent magnet) synchronous motors, as are AC synchronous motors without brushes. They are very, very, similar.

A BLDC always has armature magnets and not armature windings. 

AC synchronous motors have a field winding and can have armature windings or armature magnets. Having armature windings requires brushes though and you asked about brushless so if it is brushless, it must have magnets instead armature windings. That makes the two almost identical, only really differing in ratings and intended usage. (someone correct me if I'm wrong).

For comparison, a brushed DC motor has armature windings, while the field could be either a magnet or winding, which is the opposite which is interesting.

A BLDC motor actually does run on a form of AC when you think about it. It is just not expected to be run straight from an AC source, but a DC source with an inverter in between to synthesize the AC (though you could argue that's just a VFD and VFDs are used with AC motors all the time).

The BLDC motor is more likely to have a trapezoidal BEMF while the AC brushless motor is more likely to have a sinusoidal BEMF to optimize operation with the expected drive waveforms, but not always.