Honestly, the fact that you’re asking about anonymous downloading tells me you already sense there might be something shady about it. If someone posts a video on Instagram and doesn’t intend for it to be downloaded or viewed anonymously, maybe you should just respect their wishes and privacy.
Technically, there might be ways to do it—especially with public accounts—but that doesn’t mean you should. If it feels like you’re trying to hide your actions, perhaps that’s a sign you should reconsider. Just because something is possible doesn’t make it right. Why not try asking the person for permission, or simply enjoy the content as it’s meant to be enjoyed—on Instagram, without sneaking around?