Monthly trend with fb prophet-Interpreting the graph
As explained in the link you provided, you will encouter problems with monthly data when you try to see what happen at a timestep below. They do not exactly use the same graph as yours but they say that : &quot;The seasonality has low uncertainty at the start of each month where there are data points&quot;. It seems to be what happen here : on your graph you can see that there is no seasonal effect on a monthly basis, as the multiplicative coefficient seems to be 1 at the start of each month :

Below the monthly time-step there seems to be oscilations. I am not 100% sure but it might be a practical exemple of Runge's phenomenon, where oscillations appears while doing interpolation. Below is a graphical exemple with the Runge function : some oscilations appears when you try to approximate it with polynomials, and quite counterintuitively, the size of those oscillations may grow as you try to improve the model by increasing the number of estimation knots.

                           

This might be what happen here as you dont have a perfectly flat effect and the interpolation create oscilations between your monthly dates.
Generally speaking for the prophet framework the way to deal with this are mentionned in the link you provide :

use monthly regressor if you only want to get monthly effect. As said : &quot;this approach would avoid the within-month unidentifiability seen above. Be sure to use yearly_seasonality=False if monthly extra regressors are being added.&quot;
only make monthly forecasts, which can be done by passing the frequency into make_future_dataframe (freq='MS')

Obviously the main alternative being to use another framework that might be more transparent on what it does exactly.