Hi,
what version of AEM are you using? Please review the documentation below. Moving Tags should be available, but as mentioned by edubey not recommended in the documentation.
When a Tag is moved, all the references that point to the old tag are not changed, however there is an internal reference kept to the old tag. So this is resolved by the TagManager when the tag is called/used. By changing the title, as suggested in the documentation, no reference needs to be kept, hence cleaner. I have seen developers move a tag and then by not using the TagManager to resolve a tag, they get the old title value. So be careful and understand what happens when moving a Tag.
Regards,
Opkar
[1] https://docs.adobe.com/docs/en/aem/6-1/administer/content/tags.html#Moving Tags
[2] https://docs.adobe.com/docs/en/cq/5-6-1/wcm/tag_admin.html#Moving%20Tags
