What is cognate miRNA?
The use of the word cognate in molecular biology reflects adjectival definition 2 from Lexico:

2 formal Related; connected.
‘cognate subjects such as physics and chemistry’

The example given shows that this can be used in the most general sense of related, so that there is no requirement for some common antecedent as a surrogate for a human mother or ancestor. 

Specifically there is no implication of the involvement of a common DNA sequence, as in the answer from @Dirigible, nor that of a common origin as invoked in an earlier answer on this topic. 

This is quite clear if one considers the much earlier usage of cognate in molecular biology in relation to an mRNA codon and the tRNA anticodons that can recognize it, or a tRNA and an aminoacyl tRNA synthetase that can aminoacylate it.

As far as the usage with microRNAs is concerned, a cognate mRNA is one with which the miRNA can interact by base-pairing — no more, no less.