Am I stealing someone else's IP address by resetting my router?

No, you're not.

What your friend fails to realize is something very fundamental to
how IP addresses work.

You don't take an IP address; an IP address is
given to you.

Static IP addresses are assigned by your ISP.
Typically, it's something you need to pre-arrange as part of setting up
your account. The ISP determines what IP address to assign to you,
tells you what that IP address is, and then configures their equipment
to make sure that all data sent to that IP address is sent down the
correct wires to reach you.

For your part, when you are assigned an IP address you then need to
manually configure that in whatever equipment is connected directly to
your ISP provided internet connection. Typically, that's your router. I
have a static IP address, for example, and I had to go into my router
and configure it use that address.

Dynamic IP addresses are also assigned by your ISP,
but in a completely different manner. When you connect your computer or
router to the internet, the software asks via a protocol known as DHCP
(Dynamic Host Configuration Protocol) for an IP address. In essence, on
being connected to the net the device sends out a broadcast: "is there
anyone out there who can assign me an IP address?". The ISP's equipment
responds with "sure, here you go, you can be x.x.x.x".

On your side, you enable all this automatic IP assignment by, once
again, configuring the device connected to the internet properly. On a
router or on a computer you configure the network connection to ask for
a IP address automatically.

Two of the bigger differences between static and dynamic IP
addresses include:

• A static IP address does not change, whereas a dynamic IP address
  can. Each time a device asks for an IP address the answer could be
  different, and dynamic IP addresses are typically also only valid for
  some period of time, after which the device is required to ask
  again.

• A static IP address typically requires manual requests and
  coordination with the ISP or issuing authority. A dynamic IP address is
  assigned automatically when your computer connects.

But in all cases, static or dynamic, the IP address was given to you
by the ISP. You didn't take it. You can't "steal" what you were
given.

Either of two things happened:

1) You actually didn't have a static IP address to begin with. Since
you didn't mention configuring the router for it, I have to assume that
this is at least a possibility. Hence, resetting your router didn't
really change anything.

2) You did have a static IP address, but the reset on your router restored it to default
configuration, which is almost always to ask for a dynamic IP address.
Obviously, your ISP doesn't care, as they assigned a dynamic IP address
and you were up and running. To complete your configuration, though,
you'll want to go back into your router and configure it with the
static IP address information you should have gotten from your ISP in
the first place.