This is one of the rare occasions where I recommend consulting an expert in local labor law. This could be a lawyer but some governments, states, unions, professional associations have those too. Here is why:
I know that it's legally binding, even if I don't sign it,
Are you sure? In many legislations companies cannot unilaterally make substantial changes to the work conditions without your agreement.
Can they make an exception of the handbook just for me?
Yes. That's pretty standard practice. You will need it in writing. A local expert can tell you if any specific form is required and should review it before signing.
The new clause states that every work that I make, even if it's not for the company, belongs to the company.
I'm not sure whether you mean actual labor or IP. If it's IP this may be unenforceable . In some legislations this would be considered as too broad and non-competes and IP coverage are restricted to the "same area of business".
That doesn't let me work on my personal projects and monetize them.
Again it depends on local labor laws whether this is a legal or enforceable restrictions.
If you are lucky a simple written exception will do the trick. If the company wants to be strict about this, it will be difficult for you for you to make money on the side (even if turns out to be legal for you to do so) and it may be better looking for alternatives in the long run. I personally dislike clauses "All Your IP Belongs to US": it's not a policy of a decent employer.