Many small consequences, some with class features, others with spells.
Mechanically speaking, yes, the only consequence of them having the Fey trait is that they cannot be affected by spells and effects that target humanoids, and can be affected by spells and traits that target fey. Humanoid-targeting features are a lot to list (some that come to mind are hold/charm/dominate person, lycanthropy, the warlock's Gaze of Two Minds eldritch invocation, calm emotions, crown of madness, or simulacrum), but below is a list of all the fey-related features.
Circle of the Land druids can't be charmed by Centaurs.
Nature’s Ward
When you reach 10th level, you can’t be charmed or frightened by elementals or fey, and you are immune to poison and disease.
Oath of the Ancients paladins can turn the faithless:
Turn the Faithless. You can use your Channel Divinity to utter ancient words that are painful for fey and fiends to hear. As an action, you present your holy symbol, and each fey or fiend within 30 feet of you that can hear you must make a Wisdom saving throw. On a failed save, the creature is turned for 1 minute or until it takes damage.
They can be a Ranger's Favored Enemy (without having to pick two specific races of humanoid) and are revealed by their Primeval Awareness:
Favored Enemy
Choose a type of favored enemy: aberrations, beasts, celestials, constructs, dragons, elementals, fey, fiends, giants, monstrosities, oozes, plants, or undead.
Primeval Awareness
For 1 minute per level of the spell slot you expend, you can sense whether the following types of creatures are present within 1 mile of you (or within up to 6 miles if you are in your favored terrain): aberrations, celestials, dragons, elementals, fey, fiends, and undead.
Arcana Domain clerics' Arcane Abjuration Channel Divinity option can affect fey:
As an action, you present your holy symbol, and one celestial, elemental, fey, or fiend of your choice that is within 30 feet of you must make a Wisdom saving throw, provided that the creature can see or hear you. If the creature fails its saving throw, it is turned for 1 minute or until it takes any damage.
You might be detected by commune with nature:
You instantly gain knowledge of up to three facts of your choice about [...] powerful celestials, fey, fiends, elementals, or undead
You are detected by detect evil and good:
For the duration, you know if there is an aberration, celestial, elemental, fey, fiend, or undead within 30 feet of you, as well as where the creature is located.
You are resisted by someone affected with dispel evil and good or protection from evil and good:
For the duration, celestials, elementals, fey, fiends, and undead have disadvantage on attack rolls against you
Forbiddance might harm you:
In addition, the spell damages types of creatures that you choose when you cast it. Choose one or more of the following: celestials, elementals, fey, fiends, and undead. When a chosen creature enters the spell's area for the first time on a turn or starts its turn there, the creature takes 5d10 radiant or necrotic damage (your choice when you cast this spell).
Hallow stops you from entering, unless the caster makes an exception:
First, celestials, elementals, fey, fiends, and undead can't enter the area, nor can such creatures charm, frighten, or possess creatures within it. Any creature charmed, frightened, or possessed by such a creature is no longer charmed, frightened, or possessed upon entering the area. You can exclude one or more of those types of creatures from this effect.
Magic circle can affect you:
Choose one or more of the following types of creatures: celestials, elementals, fey, fiends, or undead. The circle affects a creature of the chosen type [...]
You can be bound to serve someone with planar binding:
With this spell, you attempt to bind a celestial, an elemental, a fey, or a fiend to your service.
You can be left out of a temple of the gods:
The temple opposes types of creatures you choose when you cast this spell. Choose one or more of the following: celestials, elementals, fey, fiends, or undead. If a creature of the chosen type attempts to enter the temple, that creature must make a Charisma saving throw. On a failed save, it can’t enter the temple for 24 hours. Even if the creature can enter the temple, the magic there hinders it; whenever it makes an attack roll, an ability check, or a saving throw inside the temple, it must roll a d4 and subtract the number rolled from the d20 roll.
Some spells force a fey back to its homeplane, but I don't think Centaurs have a different home plane, so ignore those. With a home-brewy DM, you might be able to conjure a centaur with conjure fey, or conjure woodland beings.
Best Answer
Minotaur player characters are barrel-chested humanoids.
It says it in the introduction to the playable race, and on page 18 of Guildmaster's Guide to Ravnica and page 22 of Mythic Odysseys of Theros:
So a minotaur is affected by hold person.
As you have observed, the two playable races that are not humanoids are given racial traits that explicit designate them as non-humanoids.
NPCs use statblocks
Interstingly enough, the description of the minotaur in the Monster Manual describes minotaurs like so:
Yet, their creature type is monstrosity. Why? NPCs use statblocks. The introduction to the Monster Manual says:
One of these statistics is creature type, and a monster's statblock includes all of its creature types.
PCs use playable races
In contrast with NPCs, player characters use the character creation material found in the various sourcebooks of Dungeons and Dragons. In our particular case, we can find the minotaur playable race in Guildmaster's Guide to Ravnica and Mythic Odysseys of Theros. The Races section of Theros says:
The traits described in this chapter are the traits used for player characters playing as minotaurs. Consequently, nowhere in this chapter (or the parallel chapter in Ravnica) are Minotaurs described as monstrosities, rather they are explicitly described as humanoids.
Finally, since they lack a racial trait like the Satyr and Centaur that explicitly names their creature type as anything other than humanoid, we conclude that when the book says "Minotaurs are barrel-chested humanoids", it means they are humanoids.
Humanoid is explicitly the default creature type for playable races.
In Chapter 1 of the PHB we see (pg. 11):
This is found in the Step-by-Step Character Creation chapter of the PHB. Further, we have an important rule to follow, from the Introduction of the PHB (pg. 7):
The races which break this general rule about humanoid PCs (Satyr, Centaur) have specific rules which create the exception:
Since we see nothing similar for the Minotaur playable race, we conclude Minotaur PCs are humanoids.
I can discern my PC's creature type without the Monster Manual.
After all of this, we can finally offer this compelling heuristic argument.
Heuristic 1: I don't need the Monster Manual to create my character.
Everything I need to create my player character is found in the Player's Handbook, or if I am using non-PHB player options, whatever the attendant sourcebook is. In our case, if I am playing a Minotaur, I should be able to know everything I need to know about my Minotaur player character by consulting the PHB and either Ravnica or Theros.
Heuristic 2: Ravnica and Theros do not contain NPC statblocks for the Minotaur.
The reason we know that an NPC Minotaur has creature type monstrosity is because it says so in the Monster Manual on its statblock. But Ravnica and Theros do not contain a statblock for the Minotaur.
If I am a new player creating a Minotaur player character using Theros or Ravnica, I will first encounter in the PHB the statement:
Then, as I go on to read the playable races section of Ravnica or Theros, I will never encounter the word "monstrosity" in relation to Minotaurs, only the statement:
Minotaurs may be big, hairy, and terrifying. But even Minotaurs are people who want to be held sometimes.
Thomas casts hold person.