If I assume 1sample delay for 0.1s sampling rate, the delay is Exp[-I 2 pi f T], where T is the sampling period.

This means that you expect only 36 deg phase delay at 1Hz. In reality, it's 90deg. Huge!

Also there are suspicious zeros at ~1.6Hz and ~3Hz. This may suggest that the freq counter is doing some
internal averaging like a moving average.

It would be interesting to apply a theoretical curve on the plot. It's an intellectual puzzle.