MSQ Weather

Current story-based MSQ weather (Everlasting Light, Starshowers) is actual weather, unlike weather for FATEs.

A difference here is that while you are at the point(s) in the MSQ where the weather is active, the Skywatcher will show the weather predicted for the applicable zone(s):

starshowers in the skywatcher

Image courtesy of Mikah Tayuun@Zalera.

For NG+, the real weather (as forecast without the obstruction of MSQ) weather is happening even if you have the visual effect.
You’re unable to talk to the Skywatchers during NG+ to get weather forecasts: 

talking to the skywatcher in ng+

This is during NG+ immediately after arriving to the Crystarium, during Everlasting Light.

No fish with weather conditions has been documented to be caught during the initial MSQ during Everlasting Light or Starshowers.
Testing with Fish Eyes in patch 5.0 before it was changed also indicated that Fish Eyes didn’t work when it should have for Starchaser, if Everlasting Light were not a real weather effect overriding an eligible 6-8 Clouds window.

Conversely, fish with weather conditions have been caught during MSQ weathers during NG+.