My original question was: Why is summer not capitalized like Monday and June. After some research, my question became the one in the title.
From a shallow google search, I've read that months and weekdays are proper nouns is because the names refer to a specific day of the week or month of the year. This is corroborated by the accepted answer here. However, the answer below has a higher score than the accepted answer...
This answer justifies the weekdays are proper nouns convention with the fact that "they were formed from the names of old pagan gods...". Though, that's only true for most of the days (i.e. not Sunday and Monday).
Doing another shallow google search, I found that the names of the months are also derived from the name gods. This time they were Roman.
From what I have read, the names of seasons don't have the same origin; They appear to come from older base words.
Maybe I've answered my own question, but I would like to open this discussion to everyone so there can be a common consensus.