Linear Referencing creates points with two fields holding a Route ID (that uniquely references a feature class with polylines that contain Route IDs and measures or M values in the geometry) and a single measure (a station) and creates line segments with 3 fields holding a Route ID and two measures.
The Locate Features Along Route tool can determine the Route ID and measure of a point feature class that is created using geocoding and output a table containing the Route ID and measure of the point. Run this tool twice, once with the geocoded From point and once with the geocoded To point. The From and To points should already have the same Bike Lane Segment ID associated with them before doing this and there should be two points with the same bike lane segment ID. The points should also have the intersection name field.
Uncheck the match only closest route option when running the Locate Features Along Route tool when doing this, since you can select the true matches based on the street the bike lane segment is supposed to be on and export that set to eliminate false matches.
Now since the two table have the same bike lane segment ID (your From and To points), join the two tables on that ID and export the table. Now you have a single table that has a Route ID and two measures (a From Measure and a To measure) and you can use it to create a bike lane segment with the feature class that contains Route IDs and measures or M values in the geometry by using the Make Route Event Layer tool.