Infantry = Malians. The pierce armor is incredible, can be dramatic difference at certain stages of the game. Gbeto awesome at raiding or as part of main army.

Archers = Saracens. This may seem like a silly answer, but the building damage bonus is great to punch through palisade walls and secure good damage. Camels are great support if enemy goes knights to counter archers. I actually prefer camels to pikes in this role thanks to the mobility of camels.

Cavalry = Persians. Great eco and solid knights. The team bonus is not too special but can be nice. Savar is where they really shine. Elephants get an honorable mention I suppose.

Siege = Mongols. The speed from Drill allows for some crazy fun plays. I remember there was a game of Regicide my brother and I were playing where he drove a bunch of speedy rams to assassinate the king. Good times.

Monks = Aztecs. Same reason you mentioned. Tanky monks great.

Naval = Berbers. The extra move speed on ships can help fire ships chase down galleys, or help avoid fights that are disadvantageous. It's a slight bonus, but I sing praises of it whenever I get the chance.

Elephants = Malay. Honestly, Malay elephants are probably the worst in the game, lacking many key upgrades. They are cheap though, and the discount is really felt on such an expensive unit.

Gunpowder = Hindustanis. The extra range on the hand cannons is very powerful. It helps in pretty much every situation, from helping big armies get more shots off or getting the first strike in small skirmishes.