Hello MSPro1719,
Thank you for using Apple Support Communities! I understand that not all features work in the Reminders app on your iPhone after updating to iOS 13.1.2.
Sharing a Reminders list using the Add People function is not currently working in the Reminders app, and will be resolved in a future software update. You can view more information about this feature, and what to expect when you upgrade the reminders in your iCloud account by visiting the following support page:
Get ready for the new Reminders app - Apple Support
As far as the issue you're encountering with viewing icons and creating groups in the Reminders app, what steps are you taking to get to the screen shown in your screenshot? What options do you see when you create a new Reminders list? Can you create a group following the steps here:
Use Reminders on your iPhone, iPad, or iPod touch - Apple Support
"Lists can also be organized into groups—for example, you can create a group named Work that contains all of your work-related lists:
- Tap and hold a list.
- Drag the list onto another list.
- Give the new group a name.
- Tap Create.
Tap a group's name on the Reminders main screen to show or hide the lists that it contains."
Cheers!