The answer lies in the ancestry of any species.
On detailed examination of the legless lizard, the scientists realized that it had more in common with lizards than snakes. For eg, the legless lizard has ears and remnants of their limbs and eyelids, snakes have none of that.
Now the criteria has been created in such a way that all the recognized species of plants and animals are grouped according to their common characteristics, so if 2 organisms have more in common, they are more likely to grouped “closer”, if they have less in common, then they will grouped apart.
All organisms follow this division based on having/not having common characteristics:
Kingdom > division/phylum > class > order > family > genus > species
So for example cat and dog, they both belong to the same groups down to the level of order (Animalia > chordata > mammalia > carnivora) but after that their families are different and at that level the stuff they have in common is not enough to put them in the same group so they are grouped in different families according to the criteria.
Hope this helps, feel free to ask more questions:)
Edit : switched chordata and mammalia because chordata is a phylum and mammalia is a class and I am an idiot
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