This issue has already been reported before, although MSFT bot closed it LOL (no comments regarding this):

#2493

Funny thing in this issue is if you set IsReadOnly="True" it indeed solves the date picking problem but if you are using MVVM, you won't get your data generating notification.

                <controls:DataGridTemplateColumn Header="Pick a Date" IsReadOnly="False">
                    <controls:DataGridTemplateColumn.CellEditingTemplate>
                        <DataTemplate>
                            <CalendarDatePicker Date="{Binding myDate, Mode=TwoWay}" />
                        </DataTemplate>
                    </controls:DataGridTemplateColumn.CellEditingTemplate>
                </controls:DataGridTemplateColumn>

Debugging, I got more or less to the following conclusion:
Consider: IsReadOnly="False"

1- If I click on the DatePicker while the line is not selected yet (e.g. 20/5/1999), it works allowing me to pick up a new date (4/1/2000) BUT it won't update with the binding. The UI updates but not the binding.

2- After doing 1, if you click in the same line in the DatePicker, it will try to retrieve data from the model which is still the old 20/5/1999, but when it does, the DatePicker immediately closes without user having the chance to select anything. (Paradox between model and current UI value?)

Couldn't find a way to fix this.

Maybe creating a DataGridCalendarDatePickerColumn would be a good idea as this is usually used a lot (I believe).

Thanks.