This is an interesting question. In my experience, the most important soft skills needed for PMMs are influence management, and public speaking skills.
Influence management would be getting people from outside of your department or team to work on your project. Good influence management is not just asking people to help, but making sure they understand the value of the work they're doing. If someone says they can't help, going to their manager to help with priorities needs to be done with a soft touch. Being a tyrant to get your projects done won't get you far in the long term.
Public speaking comes into play with enablement and training sessions, as well as just speaking extemporaneously about your solution, your goals, and what you're trying to accomplish. For any products I work on, I build a baseline presentation and craft the story to tell people "who, what, when, where, why, how" in an efficient manner.
Hard skills will vary from company to company, but I've always found that an ability to crunch data is highly valued by senior leaders. Taking a complex raw data set (raw research, sales data, etc.) and turning it into insights that inform your decisions is very compelling.