With respect, if you do not set up your wife's iPhone as a new device, but restore your wife's iPhone from your backup, all your settings will be installed again, including the Apple ID that has been used at the time of the backup.
And all apps and other content from Apple will be tied to your Apple ID as well. Meaning, if she is updating or downloading those apps again, she will have to put in your Apple ID information into the download dialog.
Setting an iPhone up "as new device" will give it a fresh start, no matter if it is used or brand-new.
You can share apps and other content you bought in the past, by setting up Family Sharing:
Set up Family Sharing - Apple Support