How do I fight Mongols with big axes?

The big Mongols are called "Brutes".

In the Samurai techniques section you can learn Evasion - the very first one gives you a roll which should move you out of reach.

The Moon Stance is specifically for Brutes but it is the last of the four that you get so it takes a fair bit of effort to acquire.

The heavy arrows of the Longbow can also do a lot of ranged damage. And there are also the various quick fire weapons.

The Samurai Clan armour reduces the damage you receive quite a lot.

Getting (and using) the Moon Stance would be the best method for killing these enemies. After they've attacked, or before they get the chance to attack, press Triangle repeatedly until their guard is broken, then continue to attack with either Triangle or Square.

It will also be useful to learn their moves and how they are signaled. For example, if he raises the axe above his head, he will swing down the axe in front of him. You would want to dodge with Circle. Often times, they will go from one move right into another. The best way to observe these moves is keeping a few meters of distance from them, and either jumping or dodging as they are doing their attacks (even if they are very far away).

Alternatively, you can use throwing weapons such as the kunai or sticky bombs to stagger/damage them, then attack them while they are vulnerable. It may also help to shoot them once or twice with an arrow from a farther distance.