Is there a way to get a passport stamp when traveling country to country when you do not go through customs?

I'm afraid that all you'll get in the EU is an entry and exit stamp at your arrival and departure points. You no longer need an entry visa when traveling from country to country within the EU.

Customs aren't interested in your passport, they are interested in your possessions.

But very few places in Europe will give you an entry stamp at Immigration - unless you require a visa to enter that country

Your question is not confined to the EU let alone to the Schengen countries within the EU, so the answer has to be: Depends on the border you cross. You can get a stamp at the place of arrival in Europe, no matter where that is, if you come from overseas for example. Then it gets iffy - if you move from Belgium to Holland, for example, you'd have a hard time even finding an official who has a stamp. But if you take the TGV from Paris to Geneva, you pass a border check below the arrival platform that is similar to the immigration and customs checks at the airport, so there's you chance for a Swiss stamp.

There are several countries where you'll get passport controls when travelling within the EU. (eg UK to France)

What you won't (normally) get are passport checks between countries in SCHENGEN. (eg France to Norway - which isn't in the EU)

For the purposes of tourism forget the EU exists and that Europe consists of countries called Schengen, UK, Switzerland, Poland, etc

Passports have nthing to do with customs - you're asking about Immigration. If you are going through Immigration they may stamp it or not - up to them. If they don;t you can ask- theymay co-operate or not.

If you're within Schengen - no chance - since thre are no borders to pass through and no one to stamp anything. (Going from Belgium to Germany is like going from New Jersey to Pennsylvania - who would there be to stamp anything?)

This is slightly off topic, but I was wondering whether you HAVE to get a stamp when you enter/exit Canada from the US. I do not want my passport stamped. Will they refrain from stamping it if I ask?