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Is there a theorem in complex analysis which says something along the lines that,

If two meromorphic function have same set of poles then they are same. I mean to say that, given a set of poles $M=\{z_1,z_2,\dots,z_n\}$, there is a unique meromorphic function having poles at $z_1,z_2,\dots,z_n$

If there is such theorem, I request you to mention books/online source where I can find the proof.

Shweta Aggrawal
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    See https://math.stackexchange.com/questions/906752/can-any-meromorphic-function-be-represented-as-a-product-of-zeroes-and-poles – lhf Oct 23 '18 at 12:56

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If $f$ is a meromorphic function and $g$ is an entire function without zeros, then $fg$ is a meromorphic function with exactly the same poles as $f$.

The relevant uniqueness theorem is the Weierstrass factorization theorem.

See also the Mittag-Leffler's theorem.

lhf
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