A position in company corresponds the different functions you can have in a ryű I think. I.e. shihan or shidoin, dôjô chô and so on. Titles like that name your function or position within a certain ryű. And if you leave the ryű you also leave the function and it's title behind.
dan is no title and you are not addressing someone with it. It is "Endô shihan" for instance, but never ever Endô hachidan. dan is a certificate that states your qualification in regard to the requirements of a certain ryű. (In my federation these requirements are actually written down. You can look them up and measure me by them.) The dan certificate corresponds the mokuroku or other certificates given out in koryű. So it just says, that you have acquired certain qualities. Just like a degree from a university.
Point may be that these qualities you reached don't match the requirements of a different ryű. So maybe you will want to start anew, alltough having reached a certain graduation in another ryű ...
I think I pretty much agree, but one more question I'd have would be. What if a dan grade was given with the intention of reflecting a position within a school? Rather than as a measure of having acquired certain qualities? In our specific example, ranks were given in good faith based on the level of experience of the people in question, they were given to help facilitate the creation of an organisation, with the expectation that those receiving them would then seek to acquire the qualities they were meant to represent, by doing things like adopting the founder's syllabus, attending his seminars, and generally trying to practice and work on what he teaches. In many cases that didn't happen - for reasons I shan't go into, but those reasons aren't the fault of either party, it just worked out that way - and those people have now left, which is quite understandable. Under these circumstances I'd say the rank they held doesn't reflect their abilities in the school at all, something I think they themselves would agree with. At best it could act as a rough guide to their length of time practising and teaching aikido, so it might well be fair for them to use it to promote themselves in that regard but I'm not sure what it really means.