This is explicitly not the case. Regimand runs an Empire-spanning hospitality network, Mathilde does the same with the EIC, and there will be plenty of other examples. The only structure is that they must be to the benefit of the Empire in some way. Time and time again the Vow of Poverty has been shown to have unwritten loopholes for different facets and titles a person holds.
In all seriousness, we're going to be working incredibly closely with Eike in the future, so why not make that small extra step and just take her on as an apprentice once she reaches senior apprentice, like Regimand did with Mathilde? Her inheritance of the EIC shouldn't be a problem, as its mission statement is already in-line with the Grey Order's MO.
Eike also seems to have made a very favourable impression on the dwarves in this update, and being the 'apprentice to Lady Magister Mathilde Weber' will open a lot of doors with them for her. I'd like to point out that, in all of those oaths of secrecy we swore to the dwarves, there's a structure that we keep all information that we learn to ourselves and our apprentices.
Dwarves take apprentices seriously, and Mathilde's apprentice is going to garner a lot of interest.