The US version is a bit more far-reaching but the Common Market[1] allows me to sell almost anything in any EU country.

Incidentally, Amazon is one such example. If I buy something on the German amazon.de the contract is formed with Amazon EU S.à r.l. in Luxemburg and the goods are often shipped from a Polish warehouse.

The difficulties companies face within the EU are largely language and culture related rather than legal.

[1]: https://en.m.wikipedia.org/wiki/European_Single_Market

And quite often Amazon will even ship within Europe. Many articles on Amazon UK have a note that delivery will take 2-3 days instead of 1 day, which is always a sign that they ship it from a warehouse somewhere else in Europe (mostly Germany).

And this seems to work for FBA articles as well. But they can handle that due to their own logistic network. Parcels within Europe are still rather expensive, compared to a domestic Parcel. I can still send articles much cheaper from London to Inverness (Scotland) than to Brussels, even though Brussels is closer.

But that's probably just a function of the low B2C trade across borders, which leads to a sparse delivery network.

And the EU has the equivalent:

https://en.wikipedia.org/wiki/Schengen_Area

Are those two equivalent though? As I understand the Schengen Area is about freedom of movement whereas the Commerce Clause is about, well, commerce.

No, they are not really. Schengen is mostly about border controls. For example, Switzerland is in Schengen but they are not part of the EU customs zone, so goods cannot be freely moved across the border (although there are a lot of bilateral treaties to make this easier).

But the EU has their own version of the Commerce Clause, the Common Market.