For Activity recording and other features and services to work as intended by Apple, you should each have a separate Apple ID and be signed in accordingly on your iPhone and, if you each have one, Apple Watch.
Should you wish to share eligible past and future purchases (iTunes music, apps, books, etc), you can still do so via Family Sharing (a free Apple service).
Other features of Family Sharing include a shared family calendar. shared photo album and the ability to set reminders that will alert both of you:
The following articles provide instructions for:
Creating a new Apple ID:
How to use that new Apple ID on an iPhone:
The person who sets up a new Apple ID should also first unpair their Apple Watch from their iPhone and then pair it again after the iPhone has been configured to use the new Apple ID:
Once you are set up with your own respective Apple IDs on your devices, you can also share Activity results with each other:
Share your Activity with your Apple Watch - Apple Support