We have still not finished with ‘bhavAni tvaM’. So far we considered ‘bhavAni tvaM’ as two words. But ‘bhavAni-tvaM’ can also be considered as a single word. Then it means ‘the state of being bhavAni or parAshakti ‘. The structure of the single word is something like ‘amaratvaM’ which means ’the state of being immortal’ and like ‘kavi-tvaM’ Which means ‘poetic talent’.
So the moment the devotee says ‘bhavAni tvaM’ ambaal takes it as a request for ‘bhavAni-tvaM’ and She grants the ‘bhavAni-tvaM’ to him. In other words She gives Her own status, namely the status of sAyujya with Her to him.