English grammarians had a phase in which they tried to make English grammar more closely resemble Latin grammar and predicate logic. But natural languages don’t have to follow the same systems as each other, and certainly not follow constructed languages.
“Me and Jane rode horses” has been and is correct English in many speech communities. It is not a formal register, but a language is defined by all of its speakers, not only the elite. In contrast, “me rode a horse” has never been widely accepted. They are separate constructions–there’s no reason that a natural language should be consistent across different circumstances.
Humans have a strong tendency to exclude others based on superficial differences, like language usage, and so you’ll often find strong reactions to differing registers and dialects.