Why does binomial nomenclature seem to break case rules?

Solution 1:

While it's not really a matter of English choice since it's decided by international standard (see Frank's comment above), I think the base justification was originally a philosophical statement, viz., that a species is a Form (where Form here means Platonic form or Aristotelian essence). See for instance here.

By convention, in philosophy, these were/are capitalized -- a capitalization rule that was once common in English (e.g., capitalizing Justice when it refers to what we might now call justice proper or justice itself). You can see this usage in Locke (though it also is an oddity in English capitalization).

Solution 2:

Basically, all taxa down to and including genus (but no further) are considered proper nouns, and thus are capitalized.

However, only Genus species must be written in an italic Latin script; higher taxa aren’t italicized.

Even papers written in other scripts than Latin (say, in Greek or Cyrillic, or in Chinese or Japanese) are expected to switch to Latin italics for binomials. This rule is in the standard that biologists use for these things.

It’s when you get to talking about genes that things get weird, because the rules for italics differ between animal and plant genes. Strangely enough.