Please keep in mind that the Blogs application is designed to allow for a community to Blog, not necessarily one individual (or voice). This could be where your confusion is stemming. For example, our Blog here, is based on our Pages application as it is us issuing Blogs, not the community creating multiple blogs.
Therefore, when using the Blogs application, you can categorize the Blogs which your users can create and then the user can then categorize each entry.
You would need to edit the blog to change the category as each category is specific to the blog. I'm afraid, there is no mass move tool as this
However, just doing a move, it appears that it sets this as empty so there may need to be some internal discussion here how to handle that or make it more user friendly.