When humans breed animals, they start with a dog and end with a dog. They start with a cat and end with a cat. They start with a monkey and end with a monkey. The start with a cow and end with a cow. In other words, nothing turns into anything. What they start with is what they end up with.
The followings are quotes from Aronra, who can explain this concept far more eloquently than I can.
"Evolution produces great variety via -usually subtle- changes in physical or chemical proportion. But every new species or genus, (etc.) that ever evolved was just a modified version of whatever its ancestors were, and obviously one cannot outgrow their ancestry."
"No matter how much you or your heirs may change, you obviously can’t outgrow heredity. The very concept of common ancestry is a multi-tiered and intertwined complex phylogenetic system which shows why there can’t be any distinctly separate “kinds” to begin with! At the same time, the act of speciation splits the population presenting an eventually impassable boundary between them. We often see this demonstrated live in the form of “ring species”, where different evolutionary stages exist all at once in a geographic rather than chronological distribution. Subspecies (A) may breed with subspecies (B), and (B) may breed with (C), and (C) with (D), but (A) and (D) cannot interbreed because by the time their territories overlap again, they’ve grown too distant genetically, and can’t come back. This is when we see the formation of new features, organs, or skeletal structures, each examples of new genetic “information”. What all these show is that even though a new species of perching bird (for example) is “still” a finch, it is now a different “kind” of finch, a distinct descendant species proving there is no “boundary” against macroevolution."
"So evolution,at every level- is just a matter of incremental, superficial differences being compiled atop successive layers of fundamental similarities. These layers of similarity represent taxonomic clades which encompass all the descendants of that clade, which is why birds are still dinosaurs, and humans are still apes,according to character traits definitive of each of those groups."
The issue that you are taking terms from classification methods made for human convenience and then trying to make it seem like these are distinctions that nature actually makes, rather than just humans attempting to make discussion easier on some levels. Do you understand how the quote I posted respond to your statement and how it relates to the issue I think you are having?