Considering the blue line, how do you distinguish the two black lines from the orange continuation?
For a somewhat manual approach, the TopologyBuilder could be a start.
Another approach might be the LineOnLineOverlayer + Aggregator for the counting, see attached workspace
how-to-count-per-segment.fmw
Assuming the black lines are in a different dataset or have some other way of distinguishing them from the coloured lines.
I would use a SpatialRelator with the Requestor being the segments, and the Supplier the black lines, Test to Perform would be either Touches or Intersects, and the _related_candidates are the number of black lines that touch the segment.
Depending on your input geometry you may need to use a Path Splitter on you coloured lines to break them into individual features prior to the Spatial Relator.