Whether you like it or not, people skills and company politics play a huge role in promotions in many companies, and the more junior your role the more critical role they play in promotions. And assuming you are looking for a senior level or principal level position, people skills become much more significant. If you are senior enough to start out with then you can sometimes still move up because you have an existing reputation and previous successes to show off.
My own personal experience comes into play - I joined my current company 6 years ago as a mid-level engineer, but I had 10+ years of experience and some experience managing teams. I had more experience than at least half of the team I was working with and supporting. But I lacked the people skills to work well with both my team and the other teams (which interfacing with all of the teams is a necessity of our more senior level positions).
So I took my own performance reviews to heart and learned how to better deal with the people around me. I was a Senior-level engineer within a year after getting that initial feedback and was a Principal-level engineer 4 years later and have become widely respected within my company.
But I couldn't have gotten there without the people skills. In order to move up within a company you need to be able to work well with most everyone. Sure you will not get along with everyone but you need to be able to work together with them and count on their support, and they need to be able to count on your support as they have their own jobs to do as well.
So to answer your question directly - if you feel you deserve a promotion but your management doesn't support it because your supposed lack of people skills, then you need to look at yourself and try to figure out what is holding you back. If you don't think you can address the issues that your management has with your people skills then you likely will not succeed even if you force your way into a promotion (assuming you get one at all). If you can't address those issues, then your best course of action is to probably move on. But even if you do that, then you are likely going to be held back for the same reason as other companies (although not all)