Shipping vehicle from US to Europe

The answer from Pekka covers the paper work quite well. What I want to add is regarding the shipping itself.

You already did your research that RORO is off the table but something to consider is to share a container. Look around in the common forums and communities if you can share one with another car or motorbike!

Regarding the routes, not every port is "connected" to another one directly. The more times the container has to change ship the more expensive it gets and more time it takes. Also how frequent a route gets served greatly affects the price. Spain/Portugal do not have the huge ports like Rotterdam or Hamburg which makes it more expensive/uncommon.

Pick a good shipping agency! Again ask around before you decide for one. Also make clear which kind of costs are included. Container fees, cleaning fees, receiving agent, communication fees (no joke, friends got slapped with that one) etc etc.
Depending on the package you might need to organize your own receiving agent or have an all-inclusive deal.

Once your container/car is under way have fun using vesseltracker and other pages 😀

Source: Having shipped my car from Vladivostok, Russia to Durban, South Africa and from there back to Rotterdam in the Netherlands.

Shipping vehicles (or any valuable asset) for short periods into a foreign country is always made easier by obtaining a Carnet de Passage en Douanes. This is equivalent to a passport for the vehicle and is recognised by all countries from Angola through Zimbabwe. The purpose of the document is to permit the temporary importation of valuable assets potentially liable to be sold in a country without having to provide guarantees of any duty payable on entry. The document is purchased by providing a guarantee to the issuer who is trusted to pay over any duty and fees if the owner cannot show that the item left the country. The process for non-vehicular applications is given in When travelling internationally with valuable personal effects, how do I avoid paying duty at each border crossing?

The guarantee is administered with a booklet containing multiple counterfoil pages. On entry into a country, the customs officials stamp the first removable piece and the counterfoil with an entry stamp. On exit the customs officials stamp the second removable piece and the counterfoil with an exit stamp.

The guaranteeing party will accept a claim from the customs office if the owner cannot show the item outside the country, or cannot show a double stamped counterfoil for the country show re-export.

The same process is used for expensive photography equipment, scientific instruments and anything else liable to be "forgotten" behind by visitors who are less than scrupulous in honouring temporary import permits.

This way, the importer does not need to have bags of cash to put up temporary bonds and can move through multiple countries in a way recognised as routine by customs officers who are also covered from having to accept large amounts of cash.

When the traveller returns to their home country (where they need to have paperwork in place to permit re-import to avoid duty, yet again) they can return the completed carnet to the guarantor who will release the bond.

It is important to realise that having the item stolen while in the country can trigger a demand for duty as the host country cannot necessarily trust a theft report as authoritative and may claim for the duty. Insurance should include the cost of the duty as the bond is forfeited when a claim is lodged.

European countries (Temporary admission rules linked to assure currency) are probably reasonable, but some countries demand, up to 100% of the value of the items, on entry if a carnet is not presented.

In this scenario, you would keep the vehicle on the registration of its regular country and not have to get involved with local officials.

Again, in Europe, confirm that you have the correct International Driving Permit(s) for the countries you are visiting if you don’t have an International Driving License. European countries in particular, can be members to one or more of three treaties that determine which permit(s) they accept. The documents are remarkably similar and differ primarily on their cover with the date of the treaty that the document complies with.

There are similarly multiple insurance pooling arrangements like the pan-European Green Card or its equivalent for the region being visited. Without this, many border officials impose additional insurance charges on visitors.

As Willeke notes in comments, tourist cars — vehicles that are brought into a country for a limited time for tourism, then shipped or driven out of the country — are not "imported." These limited-duration entries require paperwork and preparation (including arranging insurance that satisfies the receiving country’s minimum requirements), but don’t involve payment of duty or tax on the vehicle, or require the vehicle to be registered with the receiving country’s DMV.

EDIT March 7, 2021: as Pekka correctly notes, tourist travel with one’s own vehicle will be far easier in the EU than it will be in central or south-east Asia.

Source: reading about international vehicle travel for decades. I’ve driven and ridden many times into Canada and Mexico. In 2002, I flew my California-registered BMW motorcycle to the EU (and home, afterwards) for a summer’s riding in the Alps and Dolomites.