This paper studies the uniqueness of two non-integral finite ordered meromorphic functions with finitely many poles when they share two finite sets. Also, it provides an answer to a question posed by Gross for a particular class of meromorphic functions. Moreover, some observations are made on some results due to Sahoo and Karmakar, Acta Univ. Sapient. Math., doi:10.2478/ausm-2018-0025; Sahoo and Sarkar, Bol. Soc. Mat. Mex., doi:10.1007/s40590-019-00260-4.