You need to review your Spanish.

In Spanish ni never means without. Without is 'sin'. Church without Christ is iglesia sin Cristo. Ni means 'neither' or 'nor'. But you cannot say in Spanish iglesia ni Cristo just like in English you cannot say church nor Christ. So it is not correct Spanish at all. Neither church nor Christ is in Spanish "Ni iglesia ni Cristo". So you have to use both ni's.

So the name is Tagalog with Spanish orthography. It is not Spanish at all.