People believe that "my computer" or "my smartphone" has an Internet address, but this is a simplification of how it's really working.

The reality is that each network interface has at least one Internet address, and these should usually all be different.

An ordinary computer at home could be plugged into Ethernet and active on WiFi at the same time. The Ethernet interface may have an IPv4 address and a set of IPv6 addresses, and belong to their home LAN. The WiFi adapter and interface may have a different IPv4 address, and belongs to the same network, or some other network. The latter is called "multi-homing".

If you visit a site that reveals your "public" IP address(es), you may find that your public, routable IPv4 and/or IPv6 addresses differ from the ones actually assigned to your interfaces.

In order to be compliant with TCP/IP standards, your device always needs to respond on a "loopback" address in 127.0.0.0/8, and typically this is assigned to a "loopback" interface.

A network router does not identify with a singular IP address, but could answer to dozens, when many interface cards are installed. Linux will gladly add "alias" IPv4 addresses to most interface devices, and you'll see SLAAC or DHCPv6 working when there's a link-local and perhaps multiple routable IPv6 addresses on each interface.

The GP says that their work computer has a [public] routable IP address. But the same computer could have another interface, or even the same interface has additional addresses assigned to it, making it a member of that private 10.0.0.0/8 intranet. This detail may or may not be relevant to the services they're connecting to, in terms of authorization or presentation. It may be relevant to the network operators, but not to the end-user.

So as a rule of thumb: your device needs at least one IP address to connect to the Internet, but that address is associated with an interface rather than your device itself, and in a functional system, there are multiple addresses being used for different purposes, or held in reserve, and multiple interfaces that grant the device membership on at least one network.