In general, people fight over thrones because of the power it represents. For Japan, the tennō was not particularly powerful in the first place, but moreover lost secular power quite early in Japanese history. For most of the last 1,200 years, true political power was decoupled from the imperial title. Hence while many factions fought for power in Japan, most did not seek the throne itself (although internally branches of the imperial family jockeyed for the position). It was not worth it in the first place.
Secondly, though lacking true power, the throne - and by extension the imperial line that monopolised it - were the source of legitimacy. Japanese political landscape was extremely fragmented and fractious, and governance tended to default to a more council-like system of rule. So-called ruling clans usually controlled only a relatively small territory, no more than 25% of Japan at the peak. They derived authority over the rest of Japan from the sanction of the imperial court, which remained recognised as the legitimate government (even after it had ceased any real function).
An ambitious samurai lord might force the court to install a candidate to his liking as the next emperor. He could not reasonably make himself the emperor and benefit from the legitimisation that ancient office could provide. The tennō was not just the sovereign, but also a high priest the Japan's native religion - a position somewhat akin to the popes of Europe, in that it carried significant religious, ceremonial and symbolic functions beyond being a mere political leader. Consider how in the Investiture Controversy, Holy Roman Emperor Henry V installed anti-popes, but didn't try to become pope himself.
Overall, it is true that the title of Tennō was occupied by a single dynasty, in the same sense that only one dynasty sat on the French throne between Hugh Capet and Louis XVI. However, actual control of Japan was held by several different dynasties.
From about 810, the Fujiwara clan gained ascendancy by leveraging their marriage ties. In 866, Fujiwara no Yoshifusa was appointed to the office of regent, the first non-royal to hold that powerful post. He consolidated Fujiwara rule by
abusing his position excluding rival clans, which heretofore had acted as counterweights, from the court. The Fujiwara thus established what is known as sekkan rule, characterised by a monopoly on the offices of Sessho (regent) and Kampauku (chancellor), as well being the in-laws of the emperor.
In fact the imperial office was rather inundated by ceremonial duties, which increasingly became the court's chief concern during and after the Heian period. This was sufficiently troublesome that several ambitious emperors found it necessary to abdicate the throne before engaging in the game of
thrones politics. This created what was known as cloistered rule by the retried emperors. They were seen as the heads of the Imperial family,
chiten no kimi, i.e. the sovereign.
The Imperial-Fujiwara conflict ultimately facilitated the rise of the military samurai as a new political force. In time this military class seized power from the civil government, often (not always) establishing military rule known as the bakufu / shogunate. The shogunates originally existed in parallel with the still-extant civil government under the emperor, but expanded to the point of almost completely supplanting the court by the Edo period.
Since 1167, about six samurai clans were able to become (more or less) de facto rulers of Japan:
- The Taira, a cadet branch of the imperial family who gained power by usurping royal authority to bestow high offices and governorships on their own kin.
- The Minamoto, another branch of the imperial family who established the first shogunate after overthrowing the Taira.
- The Hōjō, stewards of the Minamoto Shogunate. Took over real power when the Direct Minamoto line suddenly died off.
- The Ashikaga, a branch of the Minamoto clan. Founded the second shogunate when Ashikaga Takauji championed the anti-Hōjō cause after the latter's dictatorial rule provoked a revolt.
- The Toyotomi, who inherited the conquests of Oda Nobunaga. They took over the Fujiwara's court positions and ruled Japan in the name of the traditional civil government.
- The Tokugawa, founders of the third shogunate after usurping power from the Toyotomi. They claimed descent from Minamoto and Fujiwara origins, but that seems to be a forgery.
One might consider Japan in this period similar to a modern constitutional monarchy, except of course real power was won on the field of battle, rather than in democratic elections.