iPhone 13 using a lot of mobile data

On the iPhone, go to Settings>Cellular (or Mobile Data depending on how your country defines it) and scroll down and you can look at the apps and services using the highest amount of cellular data and troubleshoot from there. If there are a number of things using cellular you do not want using it, you can turn off access to it there. You can also reset the statistics at the bottom of the page and let it go for 24 hours and make a comparison.

When you upgrade phones, a lot of things need to be indexed on the new device if you restored a backup from the old device. If you have it set to use cellular data, that can be used if you are not connected to wi-fi. Go ahead and do what Apple Support recommended and see what happens. Also check all of your settings for using cellular data and make sure they are what they were on the old phone.

I would seriously try to find out what iTunes is doing using that data. Are you using Apple Music, or any other iTunes service? I just purchased a song on iTunes and I've still only used 17.9MB under iTunes Media Services since the last time I reset my cellular data counter, which was September 24th.

I´m not using Apple Music nor any other iTunes service. When I am on the WiFi it does not use data for the iTunes service, but as soon as I turn mobile data on, it begins to use huge amounts of that data.

I have the same problem and when I look at the usage of the data I notice that Itunes Service is eating almost 1 GB of data every minute. I had to turn cell data off.

Again, I suggest you use the Get Support link at the top of the page here and contact Apple Support for assistance with that one.