Surely during the warranty period Volvo mandates that for the warranty to still be valid the part and installation has to go through a chain of custody of some kind? Like the part must be bought from some official vendor of Volvo parts at least?

No? By law both in EU and US if a manufacturer wants to deny warranty cover, they have to prove that whatever you did to the car has caused the issue. So in this case Volvo would need to prove that their own official part broke the car purely because it wasn't bought from them. You are allowed to work on your car yourself and it doesn't affect warranty, a lot of the things that the "right to repair" movement has been fighting for in electronics have been enshrined in law with cars for a very long time.

Another example - Volvo(and pretty much every manufacturer) says in their warranty terms that the warranty is only valid if the car is serviced annually at an authorized Volvo garage. They can say that, but the law in EU clearly states that you can have the annual service done at any vat-registered garage, and as long as "equivalent" parts are used, the manufacturer cannot deny you a warranty claim based on having the service done outside of their own approved dealership. Their own terms and conditions do not and cannot override the law.

How does it work when the customer department at Volvo denies the warranty claim? Do you escalate through small claims court? I’m not familiar with the EU. Does Volvo also have to compensate for time and court fees if so?

No, you just get a solicitor to write a letter based on an opinion from another expert. That is usually enough, companies also don't want to waste time in court.

The same law applies in US, this isn't some EU exclusive thing.

Also, this question applies in general whenever a company doesn't obey the law and your consumer rights - vast majority of the time, they do. But in case they don't you have solicitors, ombudsmen and yes, eventually, courts. It's rare that it gets that stage though.

That still sounds like a lot of work for the potential savings of a few hundred dollars on repairs and parts. It seems that anyone who could afford a new Volvo would be unlikely to take the risk. Insurance companies may after a collision since the repairs could be easily 5 figures but even then it seems dubious they’d replace major components like the engine or steering control with one of dubious origins since they’d open themselves up to liability risk.

I'm not sure what your point is. I'm just explaining that the law explicitly allows you to both

1) have the car serviced outside of the main dealership network and the manufacturer is not allowed to void your warranty due to this.

2) fit aftermarket parts, and the manufacturer is allowed to void your warranty only if they can prove that the part contributed to the fault.

The whole conversation started by me saying that with cars, you can buy original OEM parts without having to obtain the blessing from the manufacturer - with Apple(and other electronics makers) you can't.

>>That still sounds like a lot of work for the potential savings of a few hundred dollars on repairs and parts.

I don't know why you assume this is any work, you just have your car serviced elsewhere, if any warranty work comes up then you just say "I had the car serviced at a VAT-registered garage, as allowed by the statutory consumer rights". Done, works 99/100 times. Do some dealers try to wiggle out of it anyway? Sure. But dealers try to wiggle out of warranty repairs even if you have done everything "by the book" anyway. That's just what they do. In that case you also have to either start involving experts/solicitors/courts to actually make the manufacturer honour their promise to you. Whether it's worth it to you or not, is a different matter entirely.

Ah see this is the critical difference:

‘2) fit aftermarket parts, and the manufacturer is allowed to void your warranty only if they can prove that the part contributed to the fault.’

In the case of electronic products, we don’t assume things that things that look the same, perform the same.

An aftermarket iphone screen for example could even be from literally the same assembly line but perform so poorly that nobody reasonable could expect Apple to uphold their warranty on a phone with it installed. That’s why QC testing exists.

All other differences can be derived from that basis.