I am not fully understanding what it is you wish to do. What do you mean when you say you were "changing" your iPhone but "mistakenly took" a different one?
As far as adding an iPhone to an account, you simply need to sign into your account on the new iPhone. Read this support article to learn how to sign in with an Apple Account on various kinds of devices --> https://support.apple.com/111001
For an Apple Mobile device:
1. Open the Settings app.
2. Tap Sign in to your [device].
3. Enter your Apple ID email address, or use one of your additional email addresses*. Also your password,
4. If prompted, enter the six-digit verification code sent to your trusted device or phone number and complete sign in.
* https://support.apple.com/102529#alt
Additional email addresses
“These email addresses can also be used to sign in to your Apple Account.”