The way Weapon Mastery is worded feels a bit ambiguous to me.

"Your training with weapons allows you to use the mastery properties of two kinds of Simple or Martial Melee weapons of your choice, such as Greataxes and Handaxes."

In the example given, greataxes have the Cleave weapon mastery property and handaxes have the Vex weapon mastery property.

If I choose both of those two, like in the example, does that mean that I *only* get to use Cleave if I'm specifically wielding a greataxe? Or that I can use Cleave if I'm wielding any weapon with which I'm proficient and that also has the Cleave weapon property.

Intuitively, it feels like I would only be able to use Cleave if I'm wielding a greataxe, but the wording feels ambiguous enough that I wanted to ask.

I took it to mean that you choose two weapons, and have access to their weapon mastery abilities when using the chosen weapons.

It's like @TheEvilEsti said.

This article is also useful: Your Guide to Weapon Mastery in the 2024 Player's Handbook

based on https://www.dndbeyond.com/posts/1742-your-guide-to-weapon-mastery-in-the-2024-players

"The number of mastery properties you can learn"
"get access to the mastery properties for certain types of weapons"
"If you’re wielding a weapon and have learned its mastery property, you’ll be able to use that mastery property every turn when you make an attack with the weapon."

So it looks like you choose properties of weapons that you have access to, and master a max number of properties that your class/level allows.

Looks like each weapon only has 1 property: "lists a mastery property in addition to the weapon’s regular properties"

Both the Greataxe and Halbred have cleave. So if you have access to both of them and master cleave, then all weapons you have access 

I'm not sure what happens when you choose a viable mastery, but then wield a weapon with that property, but which you are not proficient with... "If the weapon meets the requirements in your class's Weapon Mastery feature, you can learn its mastery property and use it in combat." Does not explain it all to me.

Quote from CNYGamer99 >>

The way Weapon Mastery is worded feels a bit ambiguous to me.

"Your training with weapons allows you to use the mastery properties of two kinds of Simple or Martial Melee weapons of your choice, such as Greataxes and Handaxes."

In the example given, greataxes have the Cleave weapon mastery property and handaxes have the Vex weapon mastery property.

If I choose both of those two, like in the example, does that mean that I *only* get to use Cleave if I'm specifically wielding a greataxe? Or that I can use Cleave if I'm wielding any weapon with which I'm proficient and that also has the Cleave weapon property.

Intuitively, it feels like I would only be able to use Cleave if I'm wielding a greataxe, but the wording feels ambiguous enough that I wanted to ask.

 You chose two weapons and can use their mastery properties. 

Quote from loneknife >>

based on https://www.dndbeyond.com/posts/1742-your-guide-to-weapon-mastery-in-the-2024-players

"The number of mastery properties you can learn"
"get access to the mastery properties for certain types of weapons"
"If you’re wielding a weapon and have learned its mastery property, you’ll be able to use that mastery property every turn when you make an attack with the weapon."

So it looks like you choose properties of weapons that you have access to, and master a max number of properties that your class/level allows.

Looks like each weapon only has 1 property: "lists a mastery property in addition to the weapon’s regular properties"

Both the Greataxe and Halbred have cleave. So if you have access to both of them and master cleave, then all weapons you have access 

I'm not sure what happens when you choose a viable mastery, but then wield a weapon with that property, but which you are not proficient with... "If the weapon meets the requirements in your class's Weapon Mastery feature, you can learn its mastery property and use it in combat." Does not explain it all to me.

 The line that clarifies the intent is “when you make an attack with the weapon”.  That is, you can use the mastery property when you attack with the weapon that you gained the mastery for. So, if you have mastery with the the Great Axe, you can use Cleave when you attack with a Great Axe, but it doesn’t give you the ability to use Cleave when attacking with a Halberd. 

Also, note that it states "Whenever you finish a Long Rest, you can practice weapon drills and change one of those weapon choices."

This suggests that your decision is specific to the weapon itself, and not a specific mastery.

It does seem like a reasonable interpretation that you only get to use the weapon mastery property with the specific weapons you select. Otherwise, why mention the weapons at all as part of the rule?

Either way, I feel like the editors could have done a better job with that portion of the book.