It appears that when commenting, someone logged in to Blogger or Google will use the Blogger / Google account for authentication, automatically - instead of an OpenID account - even when preferring to use OpenID, and the blog permits use of OpenID.
Many blog owners permit use of OpenID, as comment authentication.
In the dashboard Settings - Posts and comments, set "Who can comment?" to "Registered User - includes OpenID".
"Registered User" should allow choice of Google or OpenID account, when the comment is being published.
But with the "Registered User" option selected, a guest, commenting on a blog, will be authenticated using a Blogger / Google account - if currently logged in to Blogger / Google.
In one case reported in Blogger Help Forum: Get Help with an Issue, a Blogger user, with an interfering cookie filter, was logged in to Blogger / Google - but was unable to use either the Google or OpenID account, when commenting.
A test comment was published - but it was anonymous.
Neither the Google, or OpenID, identity were displayed.
The immediate solution was to logout from Blogger / Google - then the OpenID account (which was WordPress hosted) became usable.
The general solution, to this problem, appears to be to use two browsers, when commenting.
- The primary browser, for Blogger / Google authenticated activity.
- The secondary browser (or possibly an Incognito / Private browser window), for anonymous and OpenID commenting activity.
I can't see any obvious benefit to having both Google and OpenID authentication as an alternate choice, in the same browser window - so maybe this is a righteous solution.