I have a bit more to add here. It’s not possible to say whether or not this is an East India Company button without it having the company initials (EIC) on the front.
The East India Company did indeed have its own navy (originally called the “Bombay Marine”) and used that design of button, but the same design continued to be used after the company was dissolved in 1874. The navy went through a series of name changes both before and after the East India Company’s interests were taken over by the Crown, but continued using the same design of button until India gained independence in 1947 and probably until about 1950 when India became a Republic.
The navy resumed its original name of “The Bombay Marine” in 1863; changed name to “Her Majesty's Indian Marine” in 1877; then “The Royal Indian Marine” in 1892; and finally “The Royal Indian Navy” in 1934 (until 1950).
The only significant change in button design during the entire period was a switch from the “Imperial crown” held by the lion to the “King’s crown” after Victoria’s death in 1901. The corrosion has made the crown unclear, but I think it’s an Imperial crown… so not later than the very early 1900s.
I’m not sure about the backmark. Usually it’s Harman & Co / Calcutta during that period but I can see a letter ‘S’ in the maker name. I have however seen the name expressed as “Harmans & Co” in some reference sources so that’s a possibility (although I’ve never seen a button with the name having an ‘S’ as “Harmans”).
