dnd 5e – What creatures copy themselves?

Jellies and Puddings.

You mentioned the black pudding in your question, and there are several variants of puddings, as well as several variants of jelly. These all have the ability Split:

Split. When a jelly that is Medium or larger is subjected to lightning or slashing damage, it splits into two new jellies if it has at least 10 hit points. Each new jelly has hit points equal to half the original jelly’s, rounded down. New jellies are one size smaller than the original jelly.

Honorable Mention: Hydra heads

Okay, so a hydra is a single creature. But its heads behave like they are individual creatures. Each gets its own attack and its own reaction. And here’s the relevant part:

At the end of its turn, it grows two heads for each of its heads that died since its last turn, unless it has taken fire damage since its last turn. The hydra regains 10 hit points for each head regrown in this way.

Each head you kill produces two new heads, which is pretty similar to what you’re looking for.

Too Slow: Any creature who can cast Simulacrum.

Currently, there is only one printed monster statblock containing the spell simulacrum, and that is Manshoon from Waterdeep: Dragon Heist. Note, this may not be quite what you’re looking for, it takes 12 hours to cast. Alternatively, the rules do give guidance to the Dungeon Master for swapping spells out, so any creature that can cast 7th level spells could be given simulacrum according to this guidance:

Spell Swaps
One way to customize an NPC spellcaster is to replace one or more of its spells. You can substitute any spell on the NPC’s spell list with a different spell of the same level from the same spell list. Swapping spells in this manner doesn’t alter an NPC’s challenge rating.

While this is a DM fiat type of ruling, it is one that is explicitly spelled out in the rules.

Simulacrum part II: Wish Boogaloo

Now, supposing a spellcaster with a 9th level spell slot were given wish by the DM in accordance with the guidance cited in the last section, then said caster could use wish to cast simulacrum as an action.