In one of my essays I wrote a sentence like "It's only and only a start", trying to emphasize. My counselor corrected it as "It's only a start". Is it wrong or inappropriate to use the phrase "only and only", especially in an essay?
In English*, that style of identical repetition for emphasis is seldom idiomatic. There are a few specific exceptions to that, but almost all of them emphasize a pair or sequence — head to head, little by little, and so forth — which is not the case here at all. Unless you're using one of those exceptions, it sounds quite foreign at best, and unintelligible at worst.
In this case, to emphasize "only" a bit more, you can either use typographic conventions like italics or bold, or put in an intensifying word like "really":
It's really only a start.
Similarly, to emphasize "start" more:
It's only just a start.
In either case, there is repetition, but it's not identical and doesn't use a conjunction.
*Perhaps Indian or South African dialects have more exceptions to this, but I would be surprised.