But it seems misleading to use the label "subjectively objective" for that phenomena. And I might be totally off track, in which case I am totally confused about what "subjectively objective" is supposed to be about.

Probability is subjective in one sense and objective in another sense. It's subjective in that the correct answer to "What's the probability of A?" depends on who is asking the question. It's objective in that the answer depends on who is asking the question only through the information she has and not, e.g., who she is. Part of the reason to call it subjectively objective is to acknowledge that critics of Bayesian epistemology/probability/statistics are correct, in part, when they complain that it's subjective. The objective part answers the criticism by pointing out that probability is subjective in a very benign sense and in precisely the sense we intuitively expect it to be. E.g. "Mary didn't know Jack had pocket aces, so in her situation thinking that she was highly likely to have the winning hand was correct."

Edit: clarified example