Strange Colpitts oscillator topology
When analyzing AC circuits one important principle is that power supplies are effectively short circuits to AC. So whether it is referenced to the positive or negative supply is not important.
The circuit basically can be reduced to the standard Colpitts as below.
As @Andy Aka mentions in the comments the circuit may be more like a Clapp/Gourier oscillator depending upon the value of C2. If it is of high value then it is merely for DC isolation rather than an important factor in the oscillator.
A Clapp oscillator is basically a Colpitts with a capacitor in series with the inductor to reduce the frequency dependence on the active elements characteristics.
C1 in the original circuit is represented by C2 in the circuit below.
As Andy mentions there are certain practical aspects missing - especially DC point stabilization. Amplitude stabilization is often not needed as they will build up until limited by the power supply voltage - that is also useful in that it ensures the device reaches saturation which is often the most efficient point.
I agree, L2 is probably to avoid operation on unintended frequencies as in addition to the mechanical resonance of the transducer there will be inductive elements in the wiring that could cause oscillation at much higher frequencies.

Diagram from Wikipedia.