Instagram Live is different from stories. When you join a live stream, your name appears immediately to the broadcaster and all viewers.
Live streams are harder to view anonymously than stories because they're broadcast in real-time. External tools typically can't display live streams in real-time.
After the live ends, if the host saves it as a video or post, you can view it anonymously through DarkViewer. Many celebrities also share their lives as stories afterward — which you can view anonymously.