Can't pair watch to a new iPhone: Pairing Failed

I got a new iPhone 13 Pro Max recently. Before I had iPhone 6s which was paired with my Apple Watch Series 2. When I migrated data from the old to the new, Apple Watch also migrated and somehow become attached to both iphones. But I used the watch only with the new one.

Then I noticed some problems when debugging watchOS apps (I'm also a developer). Workout sessions weren't always activated during debugging. So I decided to erase the watch. After that I can't pair them with the iPhone 13.

I'm constantly getting this error:

Pairing Failed

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Your iPhone couldn't maintain a wireless connection. You may be connected to an unreliable network, or you may need to move closer to your WLAN router. Try reconnecting and start again.

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Start Over

This error shows after "select wrist" step when "Connecting your Apple Watch. This may take a moment." screen appears.

Complete steps:

Set up for myself > Pair using camera > Set Up as New Apple Watch (also fails on this screen if you wait long enough) > Select Wrist and continue

I tried pairing them with my iPhone 6s again, and it always works. Also I tried removing that watch from FindMy and iCloud. May be not from every iCloud account, I have multiple apple ids.

I have very good and stable 4G connection on iPhone 13, it's my only source of Internet, I use Hotspot to transfer it to all devices, and I didn't try pairing over WiFi yet. When I try to pair with the 13, I disable the Hotspot, and this does't changes anything.

Tried rebooting both devices.

Also, "Start Pairing" button never work on the iPhone 13 Pro Max. I was able to initiate pairing only when I lock the screen and bring the phone to the watch. Sometimes pairing may be suggested randomly when the watch lie near the iPhone and I'm just browsing apps.

Also, on the iPhone 6s I had two watches in the list in Watch app, they just got different names. But it's the same watch! I just changed the name once. Now I deleted both entries from the list.