Fish Suit is not armor, so it can be worn with armor; the DM can overrule this if they don't think it makes sense
The DMG categorizes magic items (p. 133) as follows:
Types of magic items include armor, potions, scrolls, rings, rods, staffs, wands, weapons, and wondrous items.
The Fish Suit is categorized as a wondrous item in its description, not as an armor.
Fish suit
Wondrous item, Very Rare
The only limitation in the DMG is for wearing multiple items of the same kind (p. 141):
Use common sense to determine whether more than one of a given kind of magic item can be worn. A character can't normally wear more than one pair of footwear, one pair of gloves or gauntlets, one pair of bracers, one suit of armor, one item of headwear, and one cloak.
You can make exceptions; a character might be able to wear a circlet under a helmet, for example, or to layer two cloaks.
Armor and Wondrous Item are different kinds of magic item, and there is no rule forbidding wearing different kinds together, so they can be worn together. While the fish suit is donned and doffed, like an armor, it is not one, and thus does not conflict with wearing one, rules as written.
Ruling on this
Because of the donning and doffing clause, which is normally reserved for armors, it feels as if this item would have been better designed as a magical armor; that however would have meant you could not wear it with other armor usefully, and would have a negative impact on the characters AC in environments where it is needed. Other features that allow breathing in space, such as the 2nd level air bubble spell that allows breathing in such environments for 24 hours have no such downside. Forcing the players to forgo their armor to wear it will weaken them compared to these. It's possible that this item was set up as a Wondrous Item intentionally so you can combine it with armor.
The rules-as-written conclusion still feels somewhat unsatisfying, because common sense tells us you should not be able to wear two bulky outfits at the same time.
If the DM does not want to conclude these items will magically adjust in size to accommodate this, like is assumed for magic plate armor that somehow fits both a stocky dwarf and a lithe elf, we can look at some real world equivalents to inform a ruling.
In the real world, you typically wear gambesons or some other padding under chain mail or plate armor. The game assumes, these are already included with heavy armor. But Fish Suit, which from the picture looks even larger and more bulky than heavy armor, is no armor. So maybe it does not include such undergarments. That would allow at least for wearing light armor together with it, maybe supple medium armor like chain mail, if you feel it should not fit over plate.
I think it would not be unreasonable for a DM to expand the "common sense" rule and decide that wearing it conflicts with wearing armor because of the bulk it takes up on the wearers body, especially with heavier types of armor.