Depending on which RR's and what part of the country, it is called joint line use. The RR that uses the home line pays a fee to use the others trackage. In So. Calif, the BNSF and the UP have joint line coverage over the Cajon Pass from Riverside to Barstow and from Mojave to Bakersfield over the Tahachapi Loop. I don't know of a specific term except Joint Line.
It is a "Joint line" when two (or more) RRs build/buy a line where each could not justify doing it alone. They then share in maintenance costs in some manner. One may own more of the line than the other and thus pay more of the running costs, but it is a shared resource.
When one line owns the rails then another can get "Trackage Rights" to them. This is sometimes advantageous to the owning line and sometimes has to be ordered by the government for anti-trust or public good/ecconomic reasons. A fee is paid either way.
A joint line can also be two parallel, or almost parallel tracks, owned by separate companies but used as double tracks. The old Alton and Santa Fe had such an arrangement from near Coal City, Iliinois, to near Joliet , although BNSF now owns both tracks.
I have always thought that the joint line in Colorado was sort of the same.
When one RR acquires the right to use another's trackage it is purchasing "trackage rights". This is common when a RR has two tracks that are separated and wishes to move trains through using foreign tracks. Also more favorable grades can be a factor. There may be a round about connection that adds considerable mileage that is not as economical as purchasing rights to move over another line's track between those two points. The owning RR would ordinarily control all train movements although the crews would belong to the individual RR's.
Paired track arrangements are for joint use usually to the mutual benefit of both lines. The WP and SP had an arrangement over the Sierras whereby the trains of both RR's used WP tracks in one direction and SP tracks in the other. The arrangement allowed for operating over more favorable grades in both directions.
Tehachapi Loop is SP (now UP) tracks. The Santa Fe has operated over the SP's tracks there for many years. Cajon Pass was Santa Fe and UP trains operated on Santa Fe tracks in steam days and well into the diesel era. SP built its own tracks parallel to the Santa Fe during the 60's(?). Since UP now has the SP they have their own route now. I don't know what the arrangements are there now as I haven't followed that sort of thing much since the 70's.
There can be but wouldn't have to be any corporate connection between the two companies other than the trackage agreement.
The former SP line over the Cajon Pass goes to Palmdale, CA, (north) as opposed to going to the east or Barstow. The UP still has "Trackage Rights" (proper term, sorry) over the BNSF from Riverside to Barstow and visa versa. The Palmdale Cutoff isn't part of the equation as it doesn't go to the same location. Although the two sets of tracks have a cross over at a location known as Silverwood, near Summit. The main route for the UP to Vegas and Salt Lake City (LASL) is over the BNSF from Riverside to Barstow or Daggett where it cuts off onto its own tracks again.
Trackage rights can also be aquired during a merger - a third Railroad can get trackage rights during a merger - Suleski Transportation got Trackage rights during the CSX Norfolk south split up of Conrail.
Back in the 19th century the Denver & Rio Grande and the Santa Fe had a little "tiff" going through the Royal Gorge in Colorado. It got so bad that there was actually shooting going on between the various railroad construction camps! The Royal Gorge is too narrow in places for both sets of tracks so that's what led to the fight. A court finally awarded ownership to one and trackage rights to the other.