After my PhD I received the diploma with the (external and one internal) examiners final recommendation that was used to award my PhD. Because my Ph.D. wasn't graded and this recommendation is very positive, I want to share it on my LinkedIn profile. Is there anything that prevents me from doing this? I cannot really see any problems but I haven't found anyone else doing it.
I think the most significant issue is whether anyone would know (or bother to find out) what "very positive" means in the context of some specific university's usage. Moreover, for academic purposes, very soon after your Ph.D. something like this becomes either trivial and distracting (because subsequent references to your dissertation and/or your publications in general will supersede anything this honor is likely to provide) or pretentious (if subsequent references to your dissertation and/or your publications in general are nonexistent or few). On the other hand, for non-academic purposes, it might be fine to do this.
Strictly speaking, these comments would be the copyright of the examiners and/or of the university (depending on the nature of the examiners' contract with the university). Since what you are proposing constitutes publication, you should, strictly speaking, obtain permission to do so first. Unless the comments were envisaged as confidential, I suspect that the examiners will probably be happy to grant such permission without demanding any royalty, but that does not change the fact that failure to obtain permission first would be a breach of copyright.
In practice, it is unusual to publish examiners' reports on a thesis (and the only exception which I have encountered quotes exactly two words, one from each examiner). Doctoral degrees are not, in general, graded affairs: you either get the degree or you do not. If you are one of the very few people who got your thesis accepted with "no corrections" (in UK universities, the overwhelming majority of PhD candidates are required to make either "minor corrections" or "major corrections" after the first viva), then it is definitely worth mentioning that.