What is the 127.0.0.2 IP address for?

Yes:

http://en.wikipedia.org/wiki/Loopback
https://en.wikipedia.org/wiki/Localhost

IPv4 network standards reserve the entire 127.0.0.0/8 address block for loopback purposes. That means any packet sent to one of those 16,777,214 addresses (127.0.0.1 through 127.255.255.254) is looped back. IPv6 has just a single address, ::1.

Various Internet Engineering Task Force (IETF) standards reserve the IPv4 address block 127.0.0.0/8, in CIDR notation and the IPv6 address ::1 for this purpose. The most common IPv4 address used is 127.0.0.1. Commonly these loopback addresses are mapped to the hostnames, localhost or loopback.

or from the RFC itself:

RFC 3330 - Special-Use IPv4 Addresses

127.0.0.0/8 - This block is assigned for use as the Internet host loopback address. A datagram sent by a higher level protocol to an address anywhere within this block should loop back inside the host. This is ordinarily implemented using only 127.0.0.1/32 for loopback, but no addresses within this block should ever appear on any network anywhere [RFC1700, page 5].

For fun, try by pinging:

$ ping 127.127.127.127
PING 127.127.127.127 (127.127.127.127) 56(84) bytes of data.
64 bytes from 127.127.127.127: icmp_req=1 ttl=64 time=0.110 ms
64 bytes from 127.127.127.127: icmp_req=2 ttl=64 time=0.065 ms
^C
--- 127.127.127.127 ping statistics ---
2 packets transmitted, 2 received, 0% packet loss, time 999ms
rtt min/avg/max/mdev = 0.065/0.087/0.110/0.024 ms

I will answer each of your questions below, with references and examples. It is not as simple as Yes or No.

“Are all 127.x.x.x addresses restricted to the local machine?” Yes
“Are all 127.x.x.x addresses bound to the lo interface” Yes
“Are 127.x.x.x addresses routed over the network?” No
“Are all 127.x.x.x addresses the same?” No (depending on operating system)

127.0.0.0/8 - This block is assigned for use as the Internet host loopback address. A datagram sent by a higher-level protocol to an address anywhere within this block loops back inside the host. This is ordinarily implemented using only 127.0.0.1/32 for loopback. As described in [RFC1122], Section 3.2.1.3, addresses within the entire 127.0.0.0/8 block do not legitimately appear on any network anywhere. — RFC5735 Emphasis mine.

“Is 127.0.0.2 the same as 127.0.0.1?” NO According to rfc5735 it may be, but it does not have to be. This is an implementation defined behaviour. See your operating system manual. In any case the whole range is reserved, and must not be routed over a network.

While 127.0.0.1 to 127.255.255.254 are all local addresses bound to interface lo. They are not the same. You can use each address to bind a different service onto the same port. E.g 16 Million web-servers on port 80, only accessible from the local machine (If you don't run out of memory, or other resource first)

I have just set up a docker service to bind to 127.0.0.2:80. I have then added an alias to /etc/hosts. Now I can connect to it via http://myserver, but not via http://127.0.0.1 or http://localhost. However it is only available to this machine. As it is only on the lo interface.

I then set up another docker service to bind to 127.0.0.3:80, and a python service on localhost:80 and another on 127.0.0.4:80.

This may not work on all operating systems. I am using Debian(9) Gnu/Linux, Linux kernel 4.9.0-3-amd64. Some OSs may treat all addresses 127.0.0.1 → 127.255.255.254 the same. Some may only work with 127.0.0.1.

Note services such as ping will be listening on 0.0.0.0 (ipv4) so ping 127.127.127.127 will be received by the listener, because 127.127.127.127 is one of your addresses. However if a service listens on a specific address, then you need to use this specific address to connect to it (Depending on operating system used).

What is the 127.0.0.2 IP address for?

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