Cars have different tilts on the rear seats and different positioning of the isofix attachments. It sounds like your partner's car has especially sloped rear seats and probably low down isofix points. Some newer cars have the sloping seats, because they offer better safety and comfort to adult passengers, but the isofix is positioned higher to sort of straighten out the child seat position a bit.
Unfortunately the angle is not specified in the legal standard, and the problem is that if the seat is too reclined, it does affect the crash performance which is specified in the legal standard. The test bench used in the legal test is flatter than most vehicle seats, so vehicle seats which are very tilted end up with a problem here. In order to accommodate a better recline, the seat needs to be more carefully designed, which is more expensive so you don't tend to find it on these generic brands. Therefore, they tend to err on the side of having the seat be a bit too upright in order to get better crash test performance. That means the newborn head flop is a fairly common issue in cheaper/generic multi stage seats, although if the vehicle seats are particularly reclined, you can get the same issue even with the top brands unless the seat is specifically designed to work on a tilted seat.
The other common fitting issue with generic seats are that the inserts often don't fit snugly or fasten to the main cover and often both cover and insert are made of a shiny polyester type material, all of this tends to mean they easily slide around and constantly move themselves as you put the child in, which makes it really difficult to get a good, snug fit with the straps. You end up needing about four hands to hold the insert in place, hold the baby in place (esp a squirmy, curled up newborn) and tighten the straps properly. It's possible to work with it and get a good fit sometimes, but they are not easy to work with at all.
The other common issue with multi stage seats is that the straps are bulky to accommodate much older, larger children and they tend to swamp a newborn and sometimes you can't actually adjust them small enough to accommodate a newborn or the pads get in the way and prevent you from tightening it enough. Straps should be snug to the baby's body. This is just poor design - there's nothing for it except to wait until the baby gets bigger. Again there is not much in the legal standard for this, because the smallest dummy the seat is tested with is based on the size of a 6 week old baby, and the dummy is actually nearly 60cm long which is closer to the size of an average 3 month old.
That said, this seat will pass legal standard safety testing and will offer protection in a crash as long as a child fits into it properly, particularly when used in the rear facing mode. But it doesn't sound like it gives a good fit for a newborn, which a lot of multi-stage seats won't, and if the straps and inserts are hard to use then it can be difficult to ensure that it is fitted properly, meaning you might not get the crash performance it's tested for.
Long term this will likely continue to be an issue in your partner's car, so bear it in mind when choosing the next stage seat. An infant carrier which can be fitted with seatbelt or a flex type base (the Britax one recommended does both) is a good option for this stage. For the next stage you could look at seatbelt fitted extended rear facing seats like the Axkid seats, Avionaut Sky, or Britax Max Safe Pro or Safe Way M. Or if you are keen on isofix you could keep an eye out for when the Besafe izi Turn/Twist go on sale. They have a clever isofix attachment which counteracts a slope. Axkid One is also fine on sloped seats but is expensive.
The Axkid belt fitted next stage seats are not suitable from newborn. Axkid One and the Besafe seats can be purchased with a newborn insert but it's £100+ extra, so usually best to stick with an infant carrier.