Explain Like I'm Five

Mostly dropping the analogy as it falls apart quickly once you try to talk about more specific details.

How do I handle whether I want my phone number to be known to the world?

If you don't want people to be able to call you then you can block incoming calls. This is sort of the like IPv4 NAT case, people can't connect in (unless you forward ports). Or if you want to you can allow incoming calls. The choice is up to you now rather than being forced by a technical limitation.

Does my phone number ever change on its own or can I freely change it?

Generally you will be provided a "prefix" by your ISP. In v4 this would typically be a full address. In v6 there are a huge number of addresses inside this prefix. In both cases how often the prefix chances is up to your ISP, but for v6 you can chance the suffix you use inside of the prefix as often as you want.

Who has the phone book?

There are two main parts of "the phone book". There is "Who owns this address?" and "How do I get to this address?" Both of these are basically identical between IPv4 and IPv6.

For "Who owns this address?" there is a global directory of assignments. This is typically done in multiple layers.

1. The whole address space is split among reserved addresses and continental registries.
2. The continental registries distribute addresses to organizations or individuals.
3. The organizations and individuals distribute addresses to customers/devices.
4. This may continue, for example if your home internet connection is assigned multiple addresses (rare on v4, expected on v6) then you can assign addresses to devices in your house.

For "Who do I get to this address?" A protocol called BGP is used to advertise where an address is available from. So I may say "If you want to get to addresses 32 to 64 come talk to me". This is sort of like in a hotel how there are signs saying which room numbers are in which direction. When two networks are connected they share this information between them to establish a "routing table", so they know how to get to everywhere else on the internet.

This may look something like this:

1. Your laptop wants to talk to address 17. It has learned that addresses 1-8 are "local", everything else goes out your home internet connection.
2. Your residential ISP is connected to 4 different other networks (likely a Tier 1 ISP), two of these have access to address 17, but one is "closer" so your ISP sends it there.
3. This tier 1 ISP knows that a network it is connected to serves addresses 16-32. (Maybe this is your friend's residential ISP.) So it sends the traffic there.
4. Your friends residential ISP knows that addresses 16-20 are assigned to your friends house, so it sends the traffic there.
5. Your friends router knows that address 17 is their desktop, so it sends the traffic there.

Overall no single places knows how to get to every other address. But they know the best next step. So you don't know where 17 is, but you know to send it to your ISP, your ISP doesn't know where 17 is but knows that their partner tier 1 ISP knows how to get there, the tier 1 ISP doesn't know where 17 is, but knows that it belongs to your friend's ISP, your friends ISP doesn't know what device 17 is, but knows that it is in your friends house, then finally your friends home router actually knows that 17 is your friends desktop.

You can sort of imagine this like delivering mail. If I send mail in Canada that is addressed to England, Canada Post doesn't really care where exactly I am sending the letter. It just knows that it needs to forward it to Royal Mail and they will handle it from there.