Hi andrzejfromspace,
We understand you have some questions about sharing tags in Reminders lists, and we're happy to help.
"If you create a tag on a device with iOS 15 or iPadOS 15 or later, it'll show up as a tag in Reminders on your other Apple devices using iOS 15, iPadOS 15, or macOS Monterey or later. If your other devices are using an older software version, you won't see the tag unless it's part of the reminder's title or note, where you'll see the tag as plain text.
When you share a tagged Reminder with someone who's using iOS 15, iPadOS 15, or macOS Monterey or later, they'll see the tagged word, but it won't be an actual tag until they convert it to a tag. If you try to share a tagged note with someone who's using an older software version, they won't see the tag at all unless it's in the reminder's title or note, where they'll see the tag as plain text."
For more details on sharing tags, check out Organize reminders on your iPhone, iPad, or iPod touch
Are you currently using iOS 15 or iPadOS 15 on all of the devices?
Please let us know what you find, and we'll take it from there.
Have a great day.