It’s called Carom Billiards also known as “The Pocketless Pool” as the table has no holes on the table to sink any of the balls into.
Carom billiards is not considered pool as pool only consists of tables that has pockets in them. Examples of pool would be Traditional (which is the most common with six holes on the table), Snooker, and bumper pool.
According to Quora, the object of the game carom billiards “are more towards moving the object balls around the table with the cue ball in a particular fashion. For example, 3-cushion billiards demands that to score a point one must hit the cue ball into an object ball and that object ball must travel to 3 cushions of the table before striking the second object ball.
Failing to strike three cushions or the second object ball results in the end of your inning.” Though I have never played this game, I have seen a YouTube video where the object is to score points instead of sinking the ball into pockets as in traditional pool.
I have added the video below where you can see how it’s played. So, if you have played this version of billiards or have any feedback please add them to the comments section. Thanks