If I have the Addressbook enabled for both shipping and billing, I see this:

But if I disable it for billing, I see this:

Should we consider showing the address type in a site where there are multiple address types used in checkout but only a single address type supporting addressbook? Not sure if that's even a popular use case - I just happened across it in answering a Commerce Q&A question.

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Status:Active» Needs review
StatusFileSize
new656 bytes

indeed. there, we have Menu local tasks implementation for shipping and billing tabs, and i'm not sure it could be solved within menu items.

here is a quick solution : to have this info in the page title - drupal_set_title(), patch attached.