Nope, in fact kindy is not compulsory under MOE's regulations as well. Only pri sch onwards.
It's up to you to decide if you want to send. Personally, going to pre-nursery is a "good to have" not a "need to have" kind of thing.
Also, kindys are registered with MOE. For others, they should fall under MCYS if they are childcare centres (CCC). Children who go to CCCs get subsidised under MCYS for up to $300.
If they are not registered with either, then not supposed to operate as in the case of that very popular church kindy in Tampines many yrs ago.