Default taxonomy ‘post_tag’ added to CPT managing by custom role : nothing in the metabox

I guess your custom user role does not have the required capability to assign terms, which in case of the post_tag taxonomy would be edit_post.

So one thing to do would be giving it to your role—which is most likely not what you would like to do as this does not only affect assigning terms but also a number of other actions).

Another thing you could do is using the map_meta_cap function. If you choose to take on this approach and you have any problems, just say so, and I will try to help you.