Why in English name of months are capitalized but name of seasons do not?
Is there any languages that in its orthography seasons names are capitalized?
Is it related to the calender type used by its speakers? For example in Solar Hijri calender each season exactly expanded on three months, so perhaps in Persian, season names may be capitalized. (Formally Persian is written in Arabic script)
As you know month, weekday and season names often used as adverbs and adverbs are not proper nouns! Even when they are used as noun they are not proper semantically ('today', 'yesterday', etc. are more proper because they exactly refer to specific time but even this word are not capitalized). So if we accept that main use case for capitalization is indicating proper nouns, logically we should not capitalize month, weekday and season names in usual use case, even if their names etymologically derived from proper nouns.