There is certainly some overlap in the responsibilities of designers and product managers, but that overlap occurs in the form of specifics (designer) compared to bigger picture (product manager).
Product managers want to ensure that product teams have a shared understanding of the outcome they’re trying to realize and any constraints on possible solutions. Designers take that understanding and design solution options that should meet the outcome and fit those constraints.
Designers and product managers also collaborate to understand their user’s context and to make design decisions about solutions. Designers provide expertise; product managers reflect back on how the designs achieve the overall outcome.