Self-assigned IP address is generally not deadly, unless you want to go beyond the physically-connected devices. Have you tried looking for the other Mac (presuming you have enabled File Sharing?
When your Mac powers up, each interface that is working assigns itself a pseudo-random self-assigned IP address so that its can be uniquely identified. Then when you connect to a Router, the Router provides a good local IP address so quickly, most users would swear they never had a self-assigned IP address at all.
Thunderbolt bridge connection has no Router.
If you wish, you can look at your WiFi or other Internet connections, and manually assign a good local IP address in the same range.fro trouble-free operation, you should use something that is slightly outside the range being used for DHCP.
Readers are eager to answer any additional queries on this topic.