The measurement is the theoretical duct tape between the "quantum world" and the "classical world".
But there is no such thing as a "classical world", it's just a useful approximation.
And therefore, there is also no such thing as a "measurement", it's also an approximation.
(Maybe not even an approximation, but maybe more like a projection...)
My layman feeling wrt. QM, and Copenhagen school in particular, is that we're searching for too computationally simple mental models. Most other areas of physics - like GR, SR, thermodynamics - can get away with aggregating matter into points, perfect spheres, etc. because they're working in macro scale, but QM is trying to deal with the smallest bits of our reality. Now the boundary between QM and "classical physics" is one where your quantum system will interact with 10^{double digit} amount of other quantum-relevant bits. I have a feeling that searching for what constitutes "a measurement" in such scenario is missing the point, and even talking about the macro system being entangled with the test system is pretty much skipping over all the interesting bits.
Isn't that the observer becoming entangled with the measured system?
The definitions I have found always invoke the presence of a "classical system"/"observer".
But that just kicks the can down the road, because there is no well-defined definition of a "classical system" either.
(It would help tremendously if we ever measured quantum states that weren't "collapsed", but as we've never done this so far it makes most of the stochastic collapse stuff hard to justify, even if it seems intuitively like the right approach).
Bell's inequalities show that there isn't a local state that can be there.
On the other hand nobody has any clue what the quantum measurement / wave function collapse actually is. There are theories/interpretations but no truly satisfying answers in the same way as for example Newton's equations were a satisfying answer to the elliptical movement of planets, even though we later found out in the 20th century that F ~ 1/r^2 was actually an approximation.
We simply don't know, and we have no idea when shall we know.
It's either inherent randomness or just a deep hole in the whole thing (similar to the alien chess thought experiment problem). Personally I choose to believe that the theory is just incomplete because nobody can even define what a "measurement" really is, meaning in which cases what we do is a "measurement" and in which cases it is not a "measurement". I also think that this is what people like Feynman refer to when they say things like "nobody understands QM", it's actually "nobody understands the wave function collapse", the rest is just maths.