Q: Is the lightening/USB cord the same as the cord I use to charge my iPhone 5?

Yes.   Thats a good thing, saves remembering another piece if kit to carry round.

Come to think of it, mine was concealed under a cardboard slip within the box.   Look carefully in case you missed it.  

Happy to have helped.  Thank you for the solved star.   Did you get the Edit before you replied.   I'm wondering if you found the lightning cable under a similar cardboard slip as I did?.

I have a magic mouse 2 for which the cable came in the box.  But, I do not use an iPhone  ...however, I checked the article linked below in which it suggests the lightning cable can be used for both.  Comment below the third picture down.

The headlining change between the first Magic Mouse and Magic Mouse 2, of course, is the built-in rechargeable battery versus AA slots that needed replaced frequently.

Magic Mouse 2 charges using the same Lightning cable as iPhones and iPads so Apple includes the cable in the box.

https://9to5mac.com/2015/10/14/magic-mouse-2-unboxing-video/      I don't think I've mis-led you but not having the iPhone I cannot check the ports.