Registration is required

When you rent an apartment and the owner doesn't want you to register, it means it's illegal and you rent from an unreliable owner. Everybody has to register, because when you don't register you can be fined for that, but also you could be excluded from several government services. And most important: in The Netherlands you are protected by law when you hire a house or apartment, the owner can't throw you out. But since you're not registered, you don't have that protection.

The best thing you could do, is to find another place to live as soon as possible. But also, if you feel up to it, go to the town office and ask them what you could do.