All texts I read on the topic assume a metric compatible metric. Then I was thinking if this is because a non compatible connection is ill defined or something like that
If you are asking if a model with a connection that is not metric compatible is mathematically consistent, it is.
If you are asking if a model with a connection that is not metric compatible is physically reasonable, such models don't seem to have worked well so far in matching observations, but AFAIK the question is not completely closed.
If you are asking something other than the above, it should be evident that nobody understands what. So if that's the case, either you need to clarify your question further, or this thread will be closed.
The answer to your question is no, connections dont always have metric compatibility.
To expand on that, it is perfectly possible to have a connection without your manifold having a metric at all so quite clearly metric compatibility cannot be a constraint on a general connection. There are many possible meanings of ”forbidden” or ”senseless” depending on the context of the use of those words. Hence my request for specification.
A clarifying (I hope) question. If I use a different connection on a pseudo-Riemannian manifold, I have a new theory of gravity (that may or may not be physically plausible and/or consistent with experiment) right? That seems to be what @romsofia and @PeterDonis are saying in #9 and #8 respectively.
However, Sean Carroll's lecture notes seem to me to define the covariant derivative by blunt assertion: we keep defining the characteristics we'd like it to have until we've picked out a unique covariant derivative operator (and hence connection - unless I'm missing something). His notes don't really justify these characteristics in physical terms, so I kind of came away with the impression that we were picking it for mathematical convenience. I'm trying to work out if I need to re-evaluate that. Reference: https://preposterousuniverse.com/wp-content/uploads/grnotes-three.pdf, first four pages, in particular on the last one: We do not want to make [zero torsion and metric compatibility] part of the definition of a covariant derivative; they simply single out one of the many possible ones.
