Also having this issue. Been trying to google this for a few weeks so glad to find your thread. I have 14 and on trying to add the 15th I get this exact same error.
I called Apple support but the lower tier techs just kind of gave me the usual run around after confirming that the limit is 16-- mostly a waste of a couple of hours. What I was hoping for was for them to just submit my ticket to higher tier support who could get back with me if they needed to see some logs or something. However, I was just never able to make that tech support break through.
I troubleshoot I removed my 14th AirTag and tried to add it back and it wouldn't let me. It wouldn't let me add a brand new one either. Frustrating. After waiting a week, I tried adding the original 14th AirTag and it worked....but still running into the error on the 15th.
I haven't seen many people with this problem unfortunately. Considering how well these work and how many Apple must be selling, I am surprised that more people haven't been having this issue. Reddit and Macrumors responses are mostly "why do you need so many??".
I suspect (with no evidence) that this could be quickly solved by looking at some logs. I also suspect this might be a server-side limit that might be being imposed based on some flags that Apple might be detecting.