What are the best ways for a manager to find out if the team is really happy and satisfied with his management style?

There are many ways to assess team satisfaction -

1. Take feedback in Regular frequency performance reviews - In one-on-one meetings, team member may get biased in giving opinions. But, many of the times, good inputs come out of the performance review meetings.

2. Take360 Degree feedback from team members and others - This gives honest opinion of team members as feedback result will not show who commented what. This is the best way to take feedbacks on his management style.

I believe that you can always assess the satisfaction of your team by having open communication between the team members and the management, get the team involved and always ask them "If you were in my position, what will you do differently to ensure the satisfaction of all team members"

Just to have some social hours with your team and have some professional questions from

them , such as ....

=>What will you do First If assigned with Managerial Role in any other business area?

=>What ways do you think to motivate and satisfy  your team members while achienve your goals?

Answers to these questions will give you the true picture hidden in answers.

Just make use of happiness index points, e.g. in SCRUM, every week...I've asked my team to vote1-10 being10 as the highest or very happy with the current project and get the average. Then part of the weekly happiness index points, is to list down the impediments or causes of what makes them unproductive then if it's within the scope of the SCRUM lead to resolve it will be resolve otherwise inform the stakeholders/product owner of the current scenario then plan to do something about it.

Their performance based on given KPIs

I see that the team's performance is the best indicator for the tema's satisfaction. Unstatisied teams underperform while satisfied ones usually performs if the required resources are available to them.