I am able to reproduce your error using QGIS 3.2.0, but it works when using QGIS 2.18.21. There are also already a couple of bug reports regarding the size assistant, with this one being probably the most recent and accurate regarding your problem.
Having said that, there is the possibility for creating a also cartographic more appealing solution with a small workaround. Since your two attributes "type of kindergarden" and "number of children" are not directly connected it is possible to show the size of your points separate from the color for each category.
For this solution choose "Categorized" as the symbology type for your layer, then choose the column with the kindergarden type and classify your data according to your needs. Add the size scaling with the size assistant for the symbol. Now your points should have different sizes, depending on your data.
On the bottom right of the layer styling panel you select Advanced > Data-defined Size Legend and enable the collapsed or separated legend according to your needs:
Now this should be visible in the Layers panel and also in your legend.
However there we have again the problem, that the symbols for the categories are not shown, which is probably again related to the bug you described and in the legend it now looks like this:
A workaround for this problem: Duplicate the original layer (this layer can be turned off in the map), deactivate the size based styling and the Data-defined Size legend for the new layer and add this duplicated layer to the legend. Delete the category names without symbols from the original layer in the legend manually and now your legend should show your desired result: