I asked a native speaker whether these phrases were equivalent in meaning and she said no (although agreeing that they both ultimately do "point" to the same concept). I asked if 年 here is basically a measure word and she said no. Is that right?
年 is not measure word. It is noun.
那年夏天: the summer of that year
那个夏天: that summer
They both refer to a specific summer, but the first one leaves the context on the year, while the second one solely focuses on the summer. To demonstrate the difference with follow-up questions:
那年夏天特别热。It was extremely hot in the summer of that year.
北京奥运那一年。The year Olympics was held in Beijing.
那个夏天特别热。It was extremely hot in that summer.
咱们一起去海边的那个夏天。The summer when you and I went to the seaside together.
in 那年夏天 那年 is an attributive （定语）modifying or restricting the noun 夏天， it seems the phrase can be expanded to 那年的夏天， 那个夏天 is without this attributive，instead the noun 夏天 is preceded by the demonstrative pronoun 那，demonstrative pronouns 这，那 like numerals according to grammar rules require measure words，but in the case of 这 and 那 are often omitted，so although 那个年的夏天 seems correct it can be shortened to 那年（的）夏天，and although when used by itself to refer to a certain time it seems 个 should not be omitted from 那个夏天，when used attributively it often is as in 那夏天的台风