This is hard to describe properly so I attached a mod to reproduce the issue.
Caroline tells the correct day of week only on the day the game was loaded. Starting from the next day she lives in the past.

DialogueIssue.zip

It seems to work fine for me:

Does it work if you remove all mods except Content Patcher and that content pack?

Thanks for your quick response! Unfortunately removing all other mods doesn't work.
Here is a picture and a detailed log:

https://smapi.io/log/5cf9e0ac4f614e9e88add12c1c0575bd

Maybe a Linux-specific issue?

Yep, it's an issue on Linux/Mac only. It's fixed in the next SMAPI update now; thanks for reporting it!