dnd 5e – Is there such a thing as a Magically Animated Coin Monster?

Looks like a reskin (from creature to construct) of a Swarm

This monster looks to be made of tiny constructs (tiny animated objects, like coins) rather than tiny beasts

As a DM I’d be able to come up with that monster by using a swarm from the MM as a guideline, and re-skinning it.

What is a swarm?

The basic form of a swarm is a whole bunch of tiny beasts who form (collectively) a monster.

{Swarms} form as a result of some sinister or unwholesome influence … even Druids can’t charm these swarms, and their aggressiveness is borderline unnatural. (MM p. 337)

Your DM seems to have made a variation by using tiny objects that are animated, rather than tiny beasts.
Beasts are creatures; constructs are creatures. (MM, p. 6)
Tiny beasts swarm => tiny constructs/object swarm via the re-skin (a fairly simple substitution).

A swarm example (Basic Rules, p. 154; also in MM, p. 338)

Swarm of Insects
Medium swarm of Tiny beasts, unaligned {reformatted for presentation}
Armor Class 12 (natural armor) / Hit Points 22 (5d8) / Speed 20
ft., climb 20 ft.
STR DEX CON INT WIS CHA / 3 (−4) 13 (+1) 10 (+0) 1 (−5) 7 (−2) 1 (−5)
Damage Resistances bludgeoning, piercing, slashing
Condition Immunities charmed, frightened, grappled, paralyzed, petrified, prone, restrained, stunned
Senses blindsight 10 ft., assive Perception 8
Languages —
Challenge 1/2 (100 XP)
Swarm. The swarm can occupy another creature’s space and vice versa, and the swarm can move through any opening large enough for a Tiny insect. The swarm can’t regain hit points or gain temporary hit points.
Actions
Bites. Melee Weapon Attack: +3 to hit, reach 0 ft., one target in the swarm’s space. Hit: 10 (4d4) piercing damage, or 5 (2d4) piercing damage if the swarm has half of its hit points or fewer.

Mix the above template with the creature type that is Construct rather than Beast: one example is a flying sword. (An object magically animated)

Flying Sword
Small construct, unaligned
Armor Class 17 (natural armor) Hit Points 17 (5d6) Speed 0 ft., fly 50 ft. (hover)
STR DEX CON INT WIS CHA 12 (+1) 15 (+2) 11 (+0) 1 (−5) 5 (−3) 1 (−5)
Saving Throws Dex +4 Damage
And so on {cut the rest for brevity} (Basic Rules, p. 108; MM p. 20)

The swarm of coins would look a lot like a swarm of insects, except that it would be made up of coins (tiny objects animated into a swarm).

The coins should do bludgeoning damage to be thematically consistent with something like the animate objects 5th level Wizard spell. The action line will look something like:

Actions
Hits. Melee Weapon Attack: +3 to hit, reach 0 ft., one target in the swarm’s space. Hit: 10 (4d4) bludgeoning damage, or 5 (2d4) bludgeoning damage if the swarm has half of its hit points or fewer.

There you go: a reverse engineering of the monster you encountered, based on re-skinning a swarm by using the Monster Manual. The vulnerability to fire looks like an artistic flourish.