Some aspects of artifacts are specific to their context. For example, links to or from a requirement that created with reference to the artifact within a module are only associated to that module context. Links created to or from a requirement within a folder are present wherever you view the requirement. You can regard the requirement in the folder as the "real" artifact and the others as aliases (I cannot recall the terminology actually used). This allows the same requirement to be used in many different contexts with different links. I suspect the same is true for tags.

Read the below document carefully and you should know what you need to do.
https://www.ibm.com/support/knowledgecenter/SSYMRC_5.0.2/com.ibm.rational.rrm.help.doc/topics/t_reuse_artifacts_modules.html

Basically, a "base artifact" is the underlying artifact that you can reuse in modules, while a "module artifact" is only valid in the context of a module. For example, let's say you have a (base) requirement artifact 123 and it has a tag "Base". When you add it to Module 45, you will now get a module artifact 123 which has no tags. Of course you still have the base artifact 123 with tag "Base". You can add a tag "Module" to the module artifact 123, without affecting the base artifact 123. Again, you can add artifact 123 to another Module 67, and you will get a new module artifact 123 without any tags. You can then add a new tag to this new module artifact without affecting the previous two artifacts. Note that these three artifacts have the same "web Id", but different UUIDs.