Almost any organic compound is "biodegradable" given enough time, the right environment, and the right species. But the simple practical answer is no, HDPE is quite non-biodegradable. Researchers are working to identify ways to biodegrade polyethylene-based polymers, and it is possible to a degree (see this review), but naturally these polymers will take hundreds of years to break down. I also found this blog post by a packaging company that addresses the question.

Could you post a specific source of the claim? It may be that they have a more specific explanation of their claim somewhere on the site.

It is tricky to judge because different countries have different rules and standards for when you can apply the "biodegradable" label to a product. And merely claiming that the product is biodegradable without actually placing a label on the product can also be subject to different rules. So it is difficult to figure out what they mean without inspecting the source. Their claim could be based on something like 80% of the material being biodegraded within a specific timeframe, or they could be making a misleading claim based on being "technically" correct and making the claim within an under-regulated context. Or.. They could also be making a misleading and illegal claim, possibly not on purpose.