The plot shows that the series has seasonality, but the conclusion of adftest if there is no need to take the difference.
In this case, What is the best way to remove the seasonality in the model?
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ADF test has little to do with seasonality. At best it can tell you whether there is a unit root. Differencing is one of the ways to treat unit roots and mean nonstationarity. It is not a go to tool for seasonality
Depending what kind of seasonality you have simple tools such as Holt_winters filter or Census Bureau's X-13 can be used