You'll love this then - one Aikikai organization run by a Japanese shihan not recognizing a dan rank issued by another (the certificate came from hombu, just like all of them) and requiring a student to re-test.
There may be a lot of underlying reasons for that, petty as they may be.
For example, if a student studied since the beginning with Shihan 'A' and before reaching Shodan, moved (city, town or even country) because of an opportunity (work, marraige partner interest etc..) and while there for an extended time trained and tested for Shodan there with Shihan 'B' - Both Shihans belonging to Aikikai... then returns 'home' and presents the certificate to the original Shihan for recognition... there is a chance that this original Shihan make take offense that the student simply took his Shodan exam with a different Shihan for a number of reasons. Maybe the two Shihan dislike each other... who knows. It may simply be a hard pill to swallow. I can see this from so many bad movie plots "I invested years into your training and then you go and do your Shodan with someone else?"
If I was the student in that situation I would contact my first Shihan and sought permission / blessing to grade for Shodan where I was currently living before simply doing so.
Firstly, I know that if I simply show up to another dojo within our organization without a letter of introduction from our association head (not my local Sensei), our Shihan, stating my rank and to please recogonize it and allow me to further my studies etc.. then I do not get to carry my rank with me. (Nor do I care as I 'know what I know' and I know there is always much to learn so... rank hardly matters...)
I suppose even at the point where I would return to my first Shihan in the example above and he told me he would only accept it if I did my exam again with him, I would do it. No big deal, passed it once already right? Just do not expect me to pay for it again. I understand he may want to see my ability for himself, see me take the test so he can be confident in backing up that certificate himself as well as my own ability to back it up representing him in his dojo.
None of what I have suggested above 'should' matter. I was just commenting on your specific mention of that situation and how 'there may be other circumstances'.
As for the whole 'Aikikai' issue (if there is one)
Within the Aikikai (or any organization) a rank issued by them 'should' apply throughout anyone associated with the orginization. This is why they issue the passports as a means of convenience for those that wish to intermix with other dojo inside the Aikikai (and in some cases out of it) as a means of common recognition of their rank via the issuing source.
That being said, the rank does not reflect the 'skill' per se. As it has been said, that is always down to the individual and their ability, not some distant organization. This is why I understand some Dojo cho's / Shihans / Sensei's will respectfully ask you to put on that white belt and train for a few months while they guage your ability and see if you are up to their measure of Shodan.
I do not 'dislike' the Aikikai. I have no reason to. Our organization is 'affliated' I believe. Our ranks are recognized with the Aikikai and dan certificates are issued from there.
Our association organizes its Seminars, invites Shihans from Japan when the opportunity arrises. Makes trips to Japan to go and see what it like to train around there... being part of the Aikikai allows for some of the political red tape to be kept to a minimum from what I understand HOWEVER (yes, it is that big) there is still more than enough politics within the Aikikai and around it to make training 'anywhere you want' even though you are part of the Aikikai, not even remotely possible.