As we have no details of the construction, materials, potential for splashing and for water to sit on horizontal surface, and the lifespan you hope for/expect from the project, we can't really advise for your specific case. But the below should help clarify some of the relevant points to help inform your decision.
I would have thought paint alone would be a fine finish. Am I wrong?
Nope.
Although by no means an exact analogy a good starting point is thinking about what's likely used on the skirting in your bathroom, or in any typical modern bathroom in fact..... then think about the walls. Yup, "latex paint".
What we call latex paint1, although they're all pretty similar, is by no means one, homogenous product however. Obviously there are interior and exterior grades to start with, but even within those two categories they can and do vary — hence the differences in finish, wear resistance, lifespan and so forth2.
Should I finish the painted wood with lacquer (or really any film forming finish)?
If you want greater waterproofing than the paint will (typically) provide by itself, yes. Most clear overcoats will also give you greater scratch resistance which could prove a decent bonus in the long term.
The thing about "latex paint" is that while they are 'waterproof' in the sense that they're not damaged by water, they are not water-blocking. This is actually one of the key selling points of the paint type as exterior paints, they allow water that does find its way into the wood to get back out again, which is good for both the wood itself and the paint film in an exterior context.
Your contractor is probably thinking conservatively (no bad thing) and wants your project to have the best chance of enduring in good shape. But whether added protection is truly necessary is a judgment call as much as anything.
- The quality of the paint you've used is one of the two key factors.
- The material type(s) are as important or more important.
- The construction details matter.
- Your use-case is definitely a factor3.
1 Note: does not contain latex LOL
2 Somewhat reflected in differences in cost, although never discount the unfortunate reality of name-brand premiums.
3 I'd want great built-in reassurance in a vanity intended for general family use than in one that only the adults will use.