Why is a measure word not required in 一年有三百六十五天?
Some nouns need a classifier, some don't; 月(month) needs one but 天(day) doesn't. 

一年有十二個月 - 月(month) needs a classifier

一年有十二個月份 - 月份(month) needs a classifier

一年有五十二週 - 週(week) doesn't need a classifier

一年有五十二個星期 - 星期(week) needs a classifier

一年有三百六十五天 - 天(day) doesn't need a classifier

一年有四季 - 季(season) doesn't need a classifier

一年有四個季節 - 季節 (season) needs a classifier

一小時有六十分鐘 - 分鐘(minute) doesn't need a classifier

一分鐘有六十秒 - 秒(second) doesn't need a classifier

Generally, if a noun can be a classifier itself, it usually doesn't need a classifier. 週, 天, 季,  分鐘 and 秒 can all be classifier

Example: 三週工資, 兩天比賽, 一季收成, 幾分鐘路程, 幾秒時間