Question about Max Hauri Clip Clap

The smartphone is only needed to program/set up the timer. Once set, the whole thing works even when you are out of the house.

How long can the unit store measurement data?

It's just a timer with a power meter. You can set it with your mobile phone via bluetooth, then it does what you set it to do without your mobile phone. There is no connection to the internet.
But before buying, be sure to read the reviews of the app on Google.

So it must also work via the internet, or how does the connection work from abroad?

It is clearly stated in the description range 15 metres