ID check crossing the Italian-Swiss border by train

I've done this twice. Both times the train stopped in Domodossola and we were told to have our IDs ready for inspection as some guards came through the train.

On both occasions they never asked to speak to me; I'm a pale white guy who was travelling with a minimum of luggage so didn't trigger any of their instincts.

A friend of East Asian descent tells me that he was questioned on such a crossing however, and on one of the occasions when I was on the train I saw them questioning an Indian guy. So expect their random checks to be somewhat less than random.

Regardless you can expect that you may potentially be asked for ID at any time when making the crossing. If you have your passport or national ID that will be completely fine.

which information, if any, from my ID do the border police of either country record when they check it?

According to Schengen rules, travels are not recorded, especially those of EU/EEA/CH nationals. The police will only check for the validity of the document, not take note of it.

I will only carry my biometric passport and driving licence.

The European driver's license is not a valid international travel document, as it doesn't show nationality and is issued regardless of the citizenship. They will ask for passport.

But as reported by others, most of the ID checks on the Swiss border are not that random, so especially if your name sounds very Italian-ish, you will be let go on a driver's license

According to standard rules the citizens of EU it is highly recommended to carry identity card (carta identitá) or passport as the proof of your identity. Please note the ID should not cover the information that valid only inside Italy (it is usually ID card for non-citizens of Italy but residents). Probably somebody can try to travel without it but I suppose he can get a lot of problems. Anyway you can get all the information here
According to basic practice any country can perform the control at the border between EU countries but without additional stamps to your passport etc.
Based to my practice: Italy <> Switzerland - basic control at the border when guards are verifiying not everybody but some individuals with random checks. They can ask you to show the documents to verify that you have all necessary permits to come to Swtizerland and they can ask you to open the luggage to verify that you have nothing forbidden with you.
It could be control or it could be not (for example Sunday evening) so nobody can say. And it is not rare. Mainly 90% that if you will travel by train from Italy you will meet the border Swiss guards team and I suppose it is just 5% that you will meet somebody in train if you will continue from Switzerland to Germany.
But for sure it is border crossing and you should obey all the rules according to the documents, forbidden things and cash limits.
Before the transparency of banking system in Switzwerland there were a lot of Italian financial police controls at the border between Italy and Switzerland. Now probably it is over.

As Italian your passport or, if you have it, your ID card will be enough to pass the border. Your driving license is needed for driving a car but as you travel by train that should not be asked for.

As far as I know there is no requirement for travel insurance going to Switzerland but at least one Schengen country asks for it.

No other requirements for passing the internal borders.