I'd say there is a minor difference in the formal meaning, negligible for understanding of the intended meaning. Через [каждые] 5 минут - there are five minute breaks between calls. Call, talk, hang up, wait 5 minutes, then call again. Каждые 5 минут - every five minutes, regardless of each call's duration.

In practice, though, both carry a meaning of "he couldn't stop calling". And 5 minutes here merely stands for "very little time".

I'd say there is a minor difference in the formal meaning, negligible for understanding of the intended meaning. Через [каждые] 5 минут - there are five-minute breaks between calls. Call, talk, hang up, wait 5 minutes, then call again. Каждые 5 минут - every five minutes, regardless of the duration of each call.

In practice, though, both carry a meaning of "he couldn't stop calling". And 5 minutes here merely stands for "very little time".