Wounded King already did an excellent job answering your post, but let me add...
The argument, for example, for molecular studies to indicate common descent and nested heirarchies is that the only reason for genetic similarities is due to common ancestry.
The best explanation for many types of genetic similarity is, indeed, common ancestry. This is because many types of genetic sequences are non-functional, or if they are functional/coding, a range of sequence variability produces the exact same protein. I'm sure you understand that genetic sequence codes for amino acid building blocks of proteins in three base codons. There are 64 possible codon combinations that output to 22 possible results - 20 amino acids + 1 start signal + 1 stop signal.
What follows is that multiple codons can code for the same amino acid.
If convergence (or even design) was the reason for interspecies similarity of a specific gene, then these various codon possibilities would appear randomly across a phylogeny. However, the opposite is true - the trend is that codon usages are found clustered within related groups in the phylogeny.
That is, even though all of the genes in the phylogeny code for the absolutely identical protein, the genetic sequence coding for the genes are variable, and cluster within the phylogeny.
Such data
strongly suggests common ancestry, and makes convergence highly unlikely.
(And this doesn't even begin to get into shared pseudogenes, broken genes that code for nothing at all. Why do chimps and humans have the identical broken GLO gene sequence, but guinea pigs and fruit bats each have a different broken GLO gene sequence? It is because the GLO gene was broken in the common ancestor to chimps and humans, from which both inherited it, while separate distinct mutations occurred in the pigs and bats. )
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