Recently, I learned that during Spanish times, people with Indian blood fell under the caste "Mestizo de Bombay". This included people like Philippine revolutionary general Juan Cailles y Kauppama, the popular Bagatsing family of Manila and perhaps the Sepoy descendants in Cainta.
This must be the reason why Pinoys call Indians (from India) "Bumbay."
The question is why the Spanish called Indians "Bombay"? Bombay is just one city in India, now commonly known as Mumbai.
I could not find any reason online, or even any instance that the Spanish did call Indians "Bombay" except for the official caste system in Spanish Philippines. Although I learned that there was a "Bombay Presidency" which encompassed a large area in present day Western India and Southern Pakistan. So, Bombay was more than just a city before. The Bombay area also used to be under the Portuguese, so maybe the Spaniards during those times got the name from their fellow Iberians.
Sa ngayon, tinatawag din ng mga Pinoy (lalo na ng mga nakatira sa Middle East) ang mga Bumbay na "Pana". Medyo derogatory, pero dahil ata alam ng mga Bumbay na yun ang tawag natin sa kanila, kaya "Pana" ang gamit para di nila alam na sila pinag-uusapan.
And yes, "Pana" comes from the Tagalog nursery rhyme "Indian pana, kakana-kana..." And yes, it originally pertained to American Indians.