In this sense Ambar is different than Arda, which refers to the world as a physical whole.
The Elves may have equated the concept of "doom" with the inhabited world because their fëar were tied eternally with Arda, as opposed to Men, whose souls escape beyond the world upon death (see: the Gift of Men).
Furthermore, the roots MBAR and MBAR-AT might be ultimately related.
- J.R.R. Tolkien, "Words, Phrases and Passages in Various Tongues in The Lord of the Rings", in Parma Eldalamberon XVII (edited by Christopher Gilson), p. 66
- J.R.R. Tolkien, Christopher Tolkien (ed.), The Lost Road and Other Writings, Part Three: "The Etymologies", p. 372
- J.R.R. Tolkien, Christopher Tolkien, Humphrey Carpenter (ed.), The Letters of J.R.R. Tolkien, Letter 283