Backing up (2) iPhones with the same Apple ID?

I'll skip all the reasons why i'm asking this question, as it is long and boring, but I have two iPhones, one with the cell service, and an old one I have had to pull out of retirement. I have to rebuild and completely wipe the one with cell phone service, which has caused me to use my old phone (which has been pressed into service). My carrier has been a big help in switching the service between the two.

Is it possible to backup the old phone, without it overwriting the new phone? I backup to my computer, but i'm not opposed to backing up to iCloud. I am just worried how everything will be saved? Will backing up the old phone override the new phone backup? I have had to completely wipe the new one and start again, and i'm having trouble conceptualizing what will happen. For what it is worth, the two phones are named differently, and they are different models.