Yes, your position is stored in the database in case you allow the app to use your location at startup. The position, however, is stored anonymously and it is not possible to see who has performed a specific measurement. The accuracy of the location varies which means the location shown on the map can sometimes be off with several hundred metres.
The location is shown for others along with your result, however, completely anonymously.
If you do not wish for us to store your location, answer "Don't allow" to the question "BBK wants to use your current location" at startup. You can still perform a measurement, but the functionality under History will de disabled.
If you want take away the ability for Bredbandskollen to see your position altogether, you can change this setting under location services in your phone's settings.
Only measurements made via mobile broadband connections are made available to others.