C++98 was developed by an international committee, part of the International Organization for Standardization (ISO). ANSI, and other national standards bodies, are members of this committee. The international standard has been adopted by ANSI as the US national standard, and likewise by many other national standards bodies.
Earlier versions were developed commercially in the US by AT&T; I'm not sure whether or not ANSI was involved in intermediate versions (EDIT: but Erik's answer suggests that it was, briefly).
Is "ANSI C++" a good description for C++98?
I would call it "Standard C++", or "C++98", or just "C++" if it's clear that I'm talking about the current standard. While technically accurate, "ANSI C++" might confuse people who don't realise that it means the same thing, particularly people from other countries who might not know what ANSI is. I would probably get odd looks if I called it "British Standard C++", an equally accurate description.
Is "ANSI C++" a good description for subsequent versions of the C++ standard?
Same answer. The C++ standards are developed by ISO, and adopted by ANSI and other bodies.