I can’t say it’s happening to me. I’ve done two tests today; whether I long press on the notification and mark it as complete, or go into the app and mark it as complete, only today’s are the ones marked as complete (see screenshot).
Are you using the new upgraded reminders app on iOS13, or did you opt to stick with the iOS12 style reminders app? (I’m using the upgraded iOS13 reminders app.)
Are you using the same across all devices (for macOS I think the upgraded reminders is only available on Catalina)? If not, it could be an issue with using newer and older versions of reminders across devices?
What device did you initially set the recurring reminders up on?
What operating system did you initially set the recurring reminders up on?
If you set them up on iOS12 and you’ve carried them over on updating to iOS13, maybe if you delete them completely and set up as new weekly recurring reminders it might resolve your issue.
If you set them up on your Mac, maybe if you delete them completely and set up on your iPhone or iPad as new weekly recurring reminders it might resolve your issue.
Or you could try deleting them completely and set up on your iPhone or iPad regardless.