What is the advantage of matching Characters with their recommended troop type?

Generally, the only mechanical benefit is the unit-specific skills in the general's skill tree. Look for skills that specify (own retinue) and a certain unit type. Here is every unit-specific skill that is exclusive to a general's retinue:

• Diligence (Sentinel) +5 melee evasion for melee infantry
• Passion (Vanguard) +25% melee damage for all shock cavalry
• Trust (Champion) +10% armor for all spear infantry
• Judgement (Strategist) +25% ranged damage
• Resourcefulness (Strategist) enables: flaming shot (for trebuchet units)
• Nobility (Commander) +20% ranged block chance for melee cavalry

Other skills work very well for both matching and non-matching unit types:

• Composure (Sentinel, Strategist, Commander) enables: fire arrows
• Fury (Vanguard) +25 charge bonus
• Intensity (Vanguard) +25 charge speed
• Mobility (Vanguard, Champion, Commander) +25% battle running speed (only applies to infantry)

These are only a handful of skills in five different 5x3 skill trees and some aren't available until your generals rank up a few times. It can still be optimal, for example, to have 4 shock cavalry and 2 melee cavalry in a Vanguard's retinue. There's no real issue with having rainbow armies early game or late game if you're only mixing swords/spears.

What the tooltip is likely referring to is recruitment of advanced units. Like another poster mentioned, each character type has a tier 2 unit available immediately. As you unlock units through the reform tree, early on they will only be available to matching generals (non-militia archers) and late-game they will be available to everyone (jade dragons).

There are two reasons

1) Some characters get special abilities, which only affect troops of that colour. e.g. strategists (blue) get an ability, which allows archers (also blue) to shoot fire arrows. Some commanders get the ability to let their cavalry use the wedge formation.

2) Advanced troop types can only be recruited by a general of the corresponding colour. So, an elite unit of yellow cavalry can only be recruited by a yellow general. The exception here are the factions special unit, so e.g. Liu Bei can recruit the elite Yi Marksmen, despite being a yellow general, because they are the Liu Bei factions unique unit.