In general, we classify things according to how we perceive them being similar or different. Fish, sharks, and whales, for example, are quite similar. They're cylindrical swimming things that live in the water and use fins for propulsion.
Fish and trees, on the other hand, are quite different. One swims in the water, the other is practically dormant on the land.
Just Use Standard Taxonomies (Mostly)
Genetic taxonomies on Earth would actually suit your world pretty well, if your world basically has a bunch of Earth-like life on it. Things in one taxonomic group, because of the genetic similarity, tend to be suited to similar behaviors in similar environments, and therefore tend to survive that way. You don't see very many mammals flying or swimming because that's quite an evolution to get from walking on land to those places.
Of course, bats and dolphins put a kink in such classifications, but it's not hard to just call bats "birds with teats" and dolphins "weird fish".
Aliens Have Categories
When you start getting into stuff that's very alien, you'll have to use your imagination. But it often becomes easier to create overlapping groups than hierarchies once you start getting too much weirdness.
A bat might be a mammal-bird, while a snake is a land-eel-fish. You go can hog-wild if you want, or just create a couple dozen primary classifications based on Earth life that you shoehorn all your alien things into.
Just Extend Our Taxonomies
Since your world still has evolution with all its charms, the obvious taxonomy is actually the one we already use with one extra at the head:
Variant, domain, kingdom, phylum, class, order, family, genus, and species.
This "variant" rank categorizes according to which creation event the given species originates in. You might label them by date, location, or both.
If creation events happen a lot, it probably makes sense to trim several ranks off, because most things end up being unnecessary:
So you end up with a lot of variants that each have two or three ranks.
On the other hand, if only a handful of creation events have ever occurred over millions to billions of years, evolution will be the driving force behind speciation, so you'll end up with a few variants that otherwise have normal-looking taxonomies.
Re-Use of Rank Names
Your scientists will likely be split about whether to re-use common ranks (do multiple variants have an "animal" kingdom) or to use different terms for all of them.
It's more convenient to call all animal-like entities "animals", but less precise. The debate may have inadvertently caused a fire in the opposing group's printing press. I mean, it was clearly caused by magicoso->aldfey->pixies.
However, maybe the creation events spit out a lot of lifeforms, but it actually re-uses a lot of them. The hound from Psinog147 is compatible with the coyote from Barple39 and the fox from Crenatia952. In this case, you'll start getting a lot of creatures that basically are in the same family, so you'd see the classification take that into account.
Your Variants Aren't Genetically Compatible
You mentioned that each creation event has non-compatible species. If you just meant the species can't naturally produce viable offspring (like a wolf and an ostrich), I wouldn't change too much.
If different species can be entirely different forms of life, so even the genetic code is made of different proteins (or even different base elements), you'd probably want more classification ranks. Anything that's got the same base proteins is in a group together and anything with the same base elements is in a group together.
So you could potentially have Variant -> Elemental Group -> Base Proteins -> Kingdom -> Etc. At this point, the "variant" group might more academic than practically significant, and might get dropped as a taxonomy rank, instead being a category of some kind.
As Spitemaster points out in another answer, your creatures with different proteins or base elements might have some weird interactions you can use for flavor or plot points.
For example, silicon-based lifeforms could be toxic to hydro-carbon lifeforms, so non-adapted HC "predators" could end up eating an SI "prey" that ends up digesting the HC dude from the inside out.
You can mix and match all of these methods as you see fit. You can also use different methods for different kinds of life, or for different cultures in your world.
The biggest thing is to come up with a set of rules on how you want to classify things then stick to them as best you can. When it gets weird, improvise, and don't be afraid to make things that seem wonky. Real life is sometimes wonky too.