No Ponzi game condition and transversality condition are the same?
Is it right to interpret the no Ponzi game condition as a finite
  horizon version of the transversality condition?


No. The "No-Ponzi-Game" or "solvency" condition is an external constraint imposed on the individual by the market/other participants. The individual would very much like to violate it.

The Transversality condition must be satisfied in order for the individual to maximize indeed its intertemporal utility. It is an optimization condition.

So they are conceptually very different aspects of the problem.

Finally the No-ponzi-game/solvency condition is not inherently of finite horizon -it extends to the infinite horizon also.