In various dictionaries, like Lewis & Short, there are two completely separate entries for the word comparo, which otherwise appear to be identical in conjugation. How is that these two entries are considered to be different words, rather than just different meanings for the same word?
They're listed separately because their meanings (and etymologies) are very different. It's sort of like "match" in English: the meanings "pair of corresponding objects" (from Proto-Germanic) and "bit of wood used to start a fire" (from French) are different enough that it's more useful to have separate dictionary entries for them.
In this case, comparō "compare, couple" comes from compar- "equivalent", from con + par-; comparō "make ready" is an intensive form of parō (also "make ready"). Many lexicographers have thus decided to treat them as homonyms, rather than different senses of the same word, and have given them separate entries.