I think you are
Many a name I have, while my neighbour starts with a bang.
India has many languages and many names. One of its neighbours is Bangladesh.
My inmates sum up to a half-rectangle and half-circle, ink's initial.
The word has five letters ("inmates"), and the numeral 5 is kinda-sorta made up of half a rectangle (on top) and half a circle (below). The beginning of "ink" I is INDIA's initial.
With the start out of care, mirror the Arab surname, a kettle distorting.
Some ingenious people in TSL chat suggest that "care" -> "interest" whose start is I; then NDI, written in lowercase
ndi, resembles a horizontally mirrored
ibn, which is a component of some Arab surnames (it's a bit like "ben" in Hebrew or "Mac" in Gaelic or indeed -son in Germanic languages like English); and that a (lowercase)
a looks a bit like an old-fashioned kettle complete with handle.
Take out my end the very beginning, no matter if you win or lose.
The end of the word INDIA is A, the beginning of the alphabet. If you remove that then what remains is INDI, whose first ("winning") and last ("losing") letters are the same.