What I wonder about is why they never tested catching boosters with the ones they've been using all along. They know these boosters inside out, so it would be a good platform to gain experience with.
The Falcon 9 is incapable of hovering, because given the number of Merlins and their limited ability to throttle it cannot achieve a thrust/weight ratio (TWR) of 1 even on a single engine throttled to the lowest it can go. Rockets are almost entirely fuel by weight at launch, when empty they are very light. Since it has a TWR >1 when near-empty, lighting up an engine means F9 will want to go up again. So with F9 SpaceX must do a "hoverslam" to land, wherein the computer lights the engine at just the right point such that it hits relative velocity of zero right at the altitude of the landing pad (be it on ship or on land). That won't do for catching one however.
With Starship all of this was considered from the start. Raptor has better throttling capability (itself an amazing technical achievement), and of course on Super Heavy there are lots of them which is another advantage of the "many, smaller engines" approach. It means that they can effectively throttle it to just 1/33*min-throttle of max thrust. And SH is also just plain heavier construction, for good reason in an economics designed big rocket but also helpful here. Combined it is actually capable of hovering when near empty.
So I'd say they definitely have carried over some important lessons from Falcon 9.
The actual catching part might perhaps not have been very transferable, given how Falcon 9 can't roll using it's single engine unlike Starship booster, and the large difference in mass.