One name, several claims
Kurama-ryū (鞍馬流) is a name that Japanese sword tradition attaches to several things at once: to a branch of the legendary Kyōhachi-ryū said to descend from Mount Kurama north of Kyōto, to the swordsmanship the young Minamoto no Yoshitsune is said to have learned on that mountain, and to a documented school of kenjutsu that survives today. Reference works distinguish the surviving school as Shōgen Kurama-ryū (将監鞍馬流), after a formative teacher's title, though the school itself uses the plain name Kurama-ryū.
The mountain and the legend
Mount Kurama is one of the charged places of Japanese martial imagination. Medieval storytelling placed the origins of the sword arts there, in transmissions from the temple's monks and, in the older tales, from the mountain's tengu. The Kyōhachi-ryū legend makes Kurama the source from which eight lines of Kyōto swordsmanship flowed. A school bearing the mountain's name therefore inherits an enormous mythic dowry, and the honest reading is that the legend explains the name's prestige rather than the school's documented pedigree.
A sword line that wears its legend lightly: the mountain's name kept, the mountain's myths held at arm's length.
The documented line
The Kurama-ryū that can actually be followed in records is an Edo-period transmission of paired-sword kata carried by a continuous teacher line into modern Japan, where it is demonstrated at classical martial arts gatherings. Its practice is sober classical kenjutsu: two swordsmen, set forms, and the mountain's name worn as inheritance rather than as proof. The archive treats the medieval attributions as tradition and the later line as the school's verifiable history.