I get from the PHB that there is no general rule for disarming an opponent but only a specific rule for Battle Master fighters.
Can we assume that other classes (even other fighters subclasses) lack the training for such move or can we deduce that everyone can attempt a disarm without the superiority dice?
The closest thing to a general rule I can see is the sidebar in the section on Grappling/Shoving a Creature (Basic Rules p.74):
Contests in Combat
Battle often involves pitting your prowess against that of your foe. Such a challenge is represented by a contest. This section includes the most common contests that require an action in combat: grappling and shoving a creature. The DM can use these contests as models for improvising others.
It certainly seems like the intention of a section like this would be to allow you to attempt special maneuvers (such as disarming) as an action in combat via a contest if you’re not as trained as the Battlemaster, but, as always, it’s up to the DM.
As of the DMG’s release (DMG p.271), there is an officially suggested contest for disarm attempts (though, to be clear, it is still an optional rule/action that may or may not be present in any given DM’s game):
Disarm
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.