The software is essentially a 'node', doing the routing. Whenever it starts it reads the config file to see if there is a private key in there. If no, it will generate a new one for you and that is your identity on the network.

> Just perhaps with something other than IPsec underneath?

That is correct, it uses standard, but not ipsec, encryption based on public key cryptography. A host that saves its private key will thus forever have the same IP address and if it runs services you connect to them using the encryption to its public key.

A node that is configured to connect to the wider yggdrasil public network will thus be reachable on a single IP with an identity that is based on public key cryptography, even if the machine is moving from network to network or even another continent.