arcgis-desktoparcgis-prospatial-join
I am using ArcGIS Pro 2.9.0
I'm attempting to run a spatial join
I have multiple target features in the same locations, and when I run spatial join, it picks up the closest join feature twice or more. I'd like for it to pick up the next available join feature so that all my joins are unique.
Is spatial join the correct tool or should I be running another processing tool?
I recently had this issue when experimenting with a Spatial Join. My solution was to try a "real-world" scenario. I was initially trialing the Spatial Join at a large scale in a random Coordinate System. My results were similar to yours. I tried the join again in my study area with random lines and the join worked.
A potential work around if the Spatial Join still does not work, would be the following:
Perform an intersect between the lines and polygons, ensure the output type is set to Line. 
The output will be a series of lines which run through / over your polygon. Use the Feature to Point Geoprocessing Tool to create a point at the centre of your line.
Best Answer
Not really an answer, but too long for a comment...
To pick up the "next available" join feature, as you say, you would have to eliminate the already-joined features. This means that you would have to run the join one feature at a time, eliminating the joined features after each single feature has been joined.
Then after each join, append the resulting one feature to the final join feature class.
However, I assume that you'd want to be very careful about the order in which your selected the individual feature to join each time, as you'd probably want the closest joins to be done first, otherwise they may never happen at all - ie, if they were otherwise also closest (but not as close) to another feature.
So you may want to to a reverse spatial join first, to determine the distances between each join, then order the features by distance, then select each feature one-at-a-time, doing a (forward/normal) spatial join on just the one selected feature, then append the results each time (or merge all the results at the end).