Expressing dependent current source voltage using KVL
How is this possible?


(1) it's a voltage source so the voltage \$V_2 - V_1\$ is fixed by this source.

(2) it's a current controlled voltage source so the 'gain' is a transresistance - the gain has units of ohms.

Transresistance is essentially from transfer resistance.  If the controlling current were in fact the current through the source, the source would be indistinguishable from resistor.

Since the controlling current is elsewhere, the voltage across the source is proportional to a current elsewhere thus, transfer resistance.


  And is there a general formula for expressing the voltage across the
  dependent current source?


To be sure, this isn't a current source (see point 1 above), it's a voltage source controlled by a current.

If it were a dependent current source, the voltage across is not fixed by the source but, rather, by the attached circuit.  In other words, there is no general formula to express the voltage across a current source in terms of the current through.