I contacted Apple, and they had two solutions:
First. (didn’t work)
Put iPhone in recovery mode, and click on computer ‘update’. It will reinstall your iOS and should repair spotlight. Maybe will work in your case.
Second. Did work
Erase your iPhone, configure it as new, and once it is configured, then log in to your iCloud account to get back contacts, messages. Do not recover the iPhone from backup - the bug is replicated somewhere there.
It is quite painful, you have to download all the apps again, and set most of your settings again, but looks like the only solution so far.
It ask started with my switch from iPhone 11 to iPhone 13 OTA. Looks like there was some error made, which was replicated also in backup. The guy at Apple told me that he met one issue like mine, but there the option first worked just fine.