Show custom menu in category and its posts

Well it looks like you are using theme that doesn’t have custom menu supporting code inside post and category templates.
Since it’s working fine in page template you should go to Appearance -> Editor and open for editing page.php or index.php and look for wp_nav_menu() with all it’s arguments. Select and copy and insert it inside your category and page template.

That’s the basic idea but it depends how your theme is implemented. If you provide me with more details I might help you more precisely.