Even though asset publisher is set to specifically show assets from a category, blogs added through that asset publisher do not automatically have its category set. This works in 6.1.x.
STEPS TO REPRODUCE
1. Add a category in Control Panel (Control Panel>Manage Liferay>Categories)
2. Create a Page and add Asset Publisher portlet
3. Configure Asset Publisher to dynamically show entries with specified category:
"Displayed assets must match these rules:" > contains > any > of the following Categories"
4. Add new blog entry through the Asset Publisher, you won't see selected category in the bottom.
In 6.2.x this does not work when setting the display rules to "all".