What information does the router transmit/store about devices?

Question

Information Security

Hello.
What information does the router transmit / store about devices that have WI-FI enabled but do not have a password to the router.
For example, a neighbor has a router, and I have WI-FI turned on in my mobile, his router is displayed on my phone when searching for Wi-FI, and does my WI-FI see his router, provided that there was no request from my device about connection.
Thank you.

Answer 1

relatively speaking, if the router is in listening mode, then it knows about the devices on the air, but does not save any information about them anywhere. those. knows here and now, not yesterday or 5 minutes ago.

Answer 2

does my WI-FI router see it, provided that there was no connection request from my device.

There are two modes of searching for a wifi AP by a client - active and passive.
In a passive search, the client listens to the broadcast and expects to receive a beacon from the AP.
In active mode, the client sends out a probe request, which explicitly detects it on the air.
Under normal circumstances, this information is not saved.