Line density for mulitple commuters in QGIS

You could try to create a 10m Buffer around your line segments. With the respective geometry operation you should be able to then give the buffer a value according to how many lines overlap with the buffer. Then you could assign the value of the buffer to the corresponding line segment (like fill a field in the line segment according to which buffer it overlaps with).

To solve this task you can use the native Line Density algorithm that ships with the QGIS processing toolbox since version 3.12. It outputs a raster layer showcasing a density value based on a radius analysis. It is documented here.