I have exactly the same problem with the RT-AX88U Pro, works fine with Chrome. I traced the issue and it seems to be Safari stating that some of the router pages cannot be found.
Will have to wait for ASUS or Apple support to wake up to the issue and fix.
I'm sorry we are both having the problem, but I'm glad to hear I'm not the only one. I was thinking it may be caused by some sort of malware if I was the only person to report it. So maybe not.
Since I posted above I have "discovered" a number of things. I'll document (it's long) what else I've done below in case more people have this same problem.
As an experiment, I set up a completely new MacOS 15 Parallel's Virtual Machine within MacOS 15. With the completely bare new OS 15 VIRTUAL MACHINE there is NOT a problem. The router pages seem to load correctly all the time. That made me concerned that my actual MacOS 15 has some malware, or at least some corruption within the OS (I had previously tried trashing about everything I could find related to Safari in the Library files and downloaded fresh versions of Safari Technology Preview without fixing the problem - so I don't think Safari itself is corrupted.)
So then I set up a completely new MacOS 15 as a Dual Boot Volume on my Mac Studio. At first I didn't load anything into it from the previous MacOS 15 volume. When I booted into the new MacOS 15 I didn't notice any problems with the router pages. Then I used Migration Assistant to copy over the System and Network settings from the "bad" OS 15 volume, and that "broke" the "NEW" MacOS 15 volume in exactly the same way as the original volume. I either couldn't get router interface login page at all, or I could login using a Private Safari page, but then some pages wouldn't load completely or not load at all after that. So again I assumed something was corrupted in the original OS 15 volume and I had copied it over into the new OS 15 volume.
So I completely erased the 2nd OS 15 volume and created a new "3rd" OS 15 volume. Again it seemed to work fine with the router pages at first. So this time I used Migration Assistant to copy over ONLY my Apps. Then I was still able to get the router login page but I occasionally get a router page that only partially loads. When that happens I can just reload the router page and that fixes it (I was never able to do that with the original OS 15 volume).
So I'm still baffled about what is going on. I've gone into the Launch Daemons and other "helper tools" and trashed everything I can that runs in the background (thinking that loading the applications started some new background processes that are interfering with Safari, but that has not changed what happens with any of the OS 15 volumes I've tried).
Nothing except an absolutely new OS 15 install seems to work perfectly, although my 3rd volume install with just apps moved over works almost every time so far. But I'm reluctant to spend hours "restoring" the new volume to the same level as my original OS 15 volume (it takes hours getting the Mail app setup with my many Smart Mailboxes and Rules because iCloud sync of multiple non-iCloud mail accounts has never fully worked to do that - not sure its intended to), and then somewhere along the way have the 3rd OS volume just break completely like the 1st OS volume.
Bottom Line for Now
So at this point I'm inclined to give up trying to find the cause (since I can login to router perfectly with Chrome, Firefox, or MS Edge), and just go on using my original OS 15 volume. I was concerned the problem was caused by malware messing up Safari, but since I don't see any other "effects", and you also have the problem, I guess I'll ignore it for now. And hopefully it is just some really difficult to understand OS 15/Safari/Asus Router incompatibility that will eventually be fixed when one of them is changed again.