It would appear having checked MtID that they are slow on updating the app:
As of April, 2022, the following combinations are supported on mobile phones and tablets:
Google Android 7, 8, 9, 10, 11 and 12 with latest version of Google Chrome, in-app (Custom tabs) or Samsung Browser 14 internet browser.
We cannot guarantee that MitID will work with other operating systems or internet browsers than the ones listed.
Having checked the Play Store listing it also targets SDK 32 (Android 12L).
My phone is NOT rooted and I guarantee it, but since the latest update the MitID app tells me that the device is rooted every time I try to log in, and that logging in has been blocked for an hour. I can assure you the fault is with the app, not my phone. It's worked just fine until I needed to use it this morning, and I have changed or installed nothing new on my phone since the app actually worked. Fix it. EDIT: Now it's blocked my MitID completely. I hate this app so much.
Unless that is one of you then the issue definitely lies with the app not the OS and isnt something that can be mitigated or investigated until they officially update the SDK and advertise support for Android13.
If you can create an issue on the tracker post crash etc with the appropriate bug report, without this it will be difficult to address.