My understanding is that his ability only applies to Melee type units (such as swordsmen), not all units that attack via melee. Therefore it should not apply to ships. The exp bonus should still apply, though.
Exactly.
Melee is a term used by the game in two ways:
1) Melee Attack as opposed to Ranged Attack
2) Melee as a unit classification
Thus, the Zulu UA only applies to units that are included in the Melee unit classification, and not to all units that lack a ranged attack.
The game really should not use the term twice in two different ways. The Melee unit classification should be called
Hand-to-Hand or something to avoid the current confusion.
Again, the Zulu UA applies only to land units that are included in the
unit classification of 'Melee', and not to all units that attack by entering the tile of their opponent (which is also called 'Melee', though this applies to Gunpowder units in most cases and to other unit types as well). To find out what units are in what classifications, consult the Civilopedia as needed.