It's just one of many forms of a class of games called Contract Rummy. (The various rounds' goals are called contracts).
The major variation here seems to be:
The discard (or rather lack of discard): I guess by your description (since one cannot make three sets with only seven cards) that players just draw a card each turn and are not required (or ...
That set is acceptable only in Indian Rummy's three deck variant. (I'll call Indian Rummy by its name Paplu here for some better clarity.)
Two-deck Paplu and regular Rummy only has sets (same card value, different suites) and runs (same suit, sequential card values) and that move isn't legal in this format.
Three-deck Paplu allows the set type you describe,...
The way rummy figures in doesn't seem very clear from your description. It sounds similar to Pounce; in that game you have a personal tableau, common piles of cards that you're building on, and a bunch of cards you hold--although it's a stack, not a hand.
If you remember the objective of your game, that would be helpful.