This may vary by university, but is it conventional to explicitly notify or invite faculty in other departments to be a cross-appointed in yours before adding them to your faculty list?
I just discovered this week that I have am included as a cross-appointed faculty member on another department's website. No big deal, and I find it flattering really, but I was surprised that no one ever asked me outright before advertising my profile on their faculty list. Is this unusual?
No. Somebody slipped up.
Yes to what @Hegemony said.
Being cross appointed may come with dual obligations - in both departments. You don't want to be caught unawares about the expectations. In your place I would reach out to both the Chairs and seek clarification.
I had a joint appointment with another department for awhile and it involved a form with signatures from me and both of the relevant department heads. Joint appointments come out of department budgets as well. On the other hand, we also have courtesy appointments, which involves no money, forms, or service obligations but allows the connected faculty to advise grad students in that department.
As noted, this cross-appointment thing can mean different things in different institutions. In some cases it's an explicit dual-obligation situation, where salary is part-paid by each dept - and I assume you'd be aware if that was your situation!
I've also known departments who would list on the website anyone across the university who worked on something related to their field (like a "studies"-type interdisciplinary dept, think Latin American Studies, who also list on their website people in modern languages who work on South American literature, the Mexicanist in the history dept, political scientists who work on Peru, geographers who look at Patagonia, etc etc). It served to advertise to grad students where they might find advisors or committee members in other departments, and I think also just to beef up to any visitors how widely represented their field was in the institution.
At my (nonresearch) school I can think of several different forms of what could be considered cross appointments.
The first is full-time faculty whose line is divided between two departments.
The second is a full-time faculty member who is in one department but teaches a certain percentage of courses in another department.
The third is a full-time faculty member who is in one department but has an official courtesy appointment in another department.
The fourth is a full-time faculty member who is in one department and sometimes teaches class in another department. (As my school's budget situation deteriorates this is more common, since some full-time faculty members have trouble making load while some departments do not have money to hire adjunct professors.)
The fifth is a full-time administrator who has a "home" (i.e., tenure) in a department. Sometimes the faculty member will have risen through the ranks, but sometimes it is somebody who was hired as an administrator and given tenure in a department. It seems of late, these administrators never actually end up teaching in their department because they move on to another school.)
None of these, however, would happen without all involved being cognizant of the arrangement.
Some grants have a requirement to include faculty researchers from multiple disciplines. I see this a lot with our larger internal and quasi-internal grants. Somebody could be window dressing a grant application and they were on a deadline . . .
I could see my university doing this. Everyone mindlessly signs off except the faculty member who nobody bothers to inform.
My first guess would also have been that its in service to officially "allow" you to teach in a certain discipline outside of your home discipline, even if you have been doing it for years. If that isn't relevant OP, then I might guess the grant window-dressing thing, though it would be crappy to put you on a grant, even just an internal thing, without your permission.
It is most likely a screwup, perhaps only on the website.
One of my neighbor departments was getting low on numbers due to a lot of retirements, so things were looking thin on their website. They went recruiting for joint appointments to enlarge their official roster (if not their capacity). It was almost an opt-out situation. "Hey we want to give you a joint appointment at our next faculty meeting, are you in?"
See whether something similar is happening at your school.
Obviously, ghost faculty are not particularly helpful to a department's future, so the tactic didn't help in the long run. It just concealed the problem to outsiders of a couple years.