Thanks for bringing this up, I believe this exposes one more angle of how currently the blog user is implemented in the ActivityPub plugin (fyi @pfefferle).
As you noticed, this is currently not possible. Actually, the ActivityPub plugin would need to extend the Enable Mastodon Plugin to allow this. I filed https://github.com/Automattic/wordpress-activitypub/issues/800 to suggest that.
I think it’d be helpful to understand in what ways you’d like to manage the blog user and what behavior you’d expect? This is how I see it but I’d be curious for your view:
In the timeline, I suppose you’d see all posts of your blog (this is already the case when you log in with any blog user).
You could view your blog followers, but you could not follow anybody.
There are profile editing capabilities coming (see https://github.com/akirk/enable-mastodon-apps/pull/157), so this could also allow editing the blog user. I think it’s worth a discussion that through this you could accidentally rename your blog when you only wanted to rename your blog profile.
I currently don’t see how you could reply via ActivityPub (which I think is one of the motivations to actually create a real blog user?), I am not sure how to prevent this except through generating an error message.
I am marking this as “not a support question” / “resolved” since this is ActivityPub plugin territory.