Do Irisalis's clones from different skills stack?

The limit is one clone per skill, not one clone in total. Repeating the same skill does not give you a second clone from it, so the way to get more clones on the field is to use different skills, your ultimate included.

That matters because every active Iris Clone joins in when the follow-up triggers. Bloom Cluster and the Blossom Swarm talent both make all active clones fire a floral beam at the target, and each Bloomcluster beam has 15 Might. Two clones fire two beams, three fire three. A freshly summoned clone also fires one beam the moment it appears.

The catch is the reset. Recasting Iris Shot resets all of Irisalis's clones, so spamming it does not add to your stack. An Iris Shot clone lasts a fairly long 30 seconds, which gives you time to get your other skills and your ultimate down before it expires. Then cycle the dash with Whirling Blossom Rain while they are all up, which is what the kit is built around.

Not settled: Irisalis's skill names are not settled. Iris Shot is also given as Iris Shadow or Flora Fall, and Iris Shadow is also given as Whirling Blossom, which spawns up to two clones when it hits. So exactly which skills count as "different" for stacking depends on which naming your version uses.