I would say neither the Cubs nor the Astros achieved Dynasty status. The late 90s in the 2000s Yankees were a dynasty. Maybe the middle 00s Red Sox could be called one. The Giants probably qualify but they also missed the playoffs in those in between years. So that hurts it some, but I'd still call them one.
For me, a dynasty team has to win multiple titles with the same core and otherwise be competitive over a 4-6 year period. The Astros still have the makings of one. The Cubs might get back in there. And a core, imo, can be around 8 players.