I want to hit enemy's axe to break it. How this is properly played in D&D 5e? And what about shield or armor? Can I attack these items to break it?
Called shots are not part of the normal rules, so the DM may not allow you to do this. However, there is some guidance for this if it is allowed.
First of all, you can disarm someone by targeting their weapon. If successful, they let go of the weapon, though it is not broken. Note this is a variant rule which needs the DM's approval to be used.
Disarm (DMG 271)
A creature can use a weapon attack to knock a weapon or another item from a target's grasp. The attacker makes an attack roll contested by the target's Strength (Athletics) check or Dexterity (Acrobatics) check. If the attacker wins the contest, the attack causes no damage or other ill effect, but the defender drops the item. The attacker has disadvantage on its attack roll if the target is holding the item with two or more hands. The target has advantage on its ability check if it is larger than the attacking creature, or disadvantage if it is smaller
This is a strange contest. It's an ability check vs an attack roll. But this is the closest you will be getting to targeting a specific weapon on their body.
In any case, you cannot make a called shot without the DM's approval, even to specific items. If the DM does approve, you should discuss with him the rules/mechanics you will use for this.
Now, about the HP of the objects to break them, should you be allowed to hit them, there is guidance there from the DMG.
Statistics for Objects (DMG 246)
When time is a factor, you can assign an Armor Class and hit points to a destructible object. You can also give it immunities, resistances, and vulnerabilities to specific types of damage.
The AC and HP of various items are then presented in table format. AC is determined by the material of the object (the weakest is cloth at AC 11, the toughest is adamantine at AC 23).
From this table, wood has AC 15 -- which covers wands, quarterstaves, polearms -- while iron/steel has an AC 19 -- which covers swords, shields, armor, and guns.
The HP of items are also given. A Tiny item is either a bottle or a lock, while a Small item is either a chest or a lute. The weapon size will likely be one of these. For Tiny objects, fragile objects have 2 (1d4) HP whereas resilient objects have 5 (2d4) HP. For Small objects, fragile ones have 3 (1d6) HP whereas resilient ones have 10 (3d6) HP.