why the name "bay of bengal " was given to the sea?

The Bay of Bengal is the largest bay in the world, forms the northeastern part of the Indian Ocean. It resembles a triangle in shape, and is bordered mostly by India to the west, Bangladesh to the north and and Burma (Myanmar) and the Andaman and Nicobar Islands to the east. It is enveloped by Indian Territory and Calcutta being the capital of India in West Bengal during the British Rule, the Bay is thus named 'Bay of Bengal'.

In English, "bay" can be defined as an indentation of the shoreline,

"larger than a cove but smaller than a gulf".

Well, the British made a mistake here, didn't they!

Obviously the Bay of Bengal is somewhat larger than the Gulf of Mexico,

much larger than the Gulf of Aden, and ten times the size of the Bay of Biscay.

These terms (cove, bay, gulf, sea) are rather nebulous, always.

You wouldn't believe some of the tiny water bodies that carry the name "sea,"

for example the Salton Sea in California, whose extent is less than 6000 km^2,

only 1/3 of 1% the size of the "Bay" of Bengal.

Maybe they picked "Bay of Bengal" because they liked the alliteration?