That he didn't even name the 10 generations of Aizu's (non-existant) Yagyu Shinkage-ryu swordsman including his own teacher (that would be quite an insult, actually, were such a man to have existed) is further evidence that this is a useful "fiction."
This is a very subtle but important detail. I don't know the specifics of this case, or how closely the traditional Japanese martial art system resembles that of the Chinese system, but in Chinese martial art, when this kind of things happen, these are the usual causes:
In feudal China, martial art was like any other type of craftsmanship, given all the normal socioeconomic incentives, important parts of the skill are only passed down to truly trusted students who will be 100% loyal to the teacher. That system is patterned after the traditional family system. In the family system, each member has clearly defined rights and duties. One of the most import duties of the child/student is to always honor and support his father. So within each martial art group, the No. 1 prohibition is 'qi shi mie zu' - to deceive, humiliate, brought shames to, ally oneself with external enemies of, or harm one's father/teacher and other ancestors. Denying their existence (I did not study with X), telling lies about who your immediate ancestors are (I studied with actual person Y) falls under this.
A student's right to ask questions is limited, he cannot ask who his teacher's teacher is, as that is considered rude - you're being skeptical about your teacher's credential. But normally, when a teacher formally accept a student as a disciple, starting the father/son relationship, one of the first things he would do is volunteer the lineage information to the best of his knowledge. Just as with real families, he may not know the names of his distant ancestors, but he should know the previous 2 generations. Under the family system, a teacher can withhold information, but he should never deceive his students.
When someone does not reveal who his teacher is, it's usually for the following reasons:
1. He was not a formal disciple.
There are innocent and interesting exceptions here: in China, if you become blood brothers with someone, you can show him everything you know, and he can even learn from your teacher. Because the basic tenant of blood brotherhood is "there is no separation between us, what is mine is yours, and what is yours is mine. Your father is now my father too, I will honor him like a real son..." In theory then, all your teachers and students are his as well, and vice versa. Under this type of circumstances, someone can become an expert without formally becoming a formal disciple.
2. He was a disciple but the relationship with teacher soured. As with real families, relationship can change over time, and the teacher may have severed the relationship later.
3. He created the style himself. In traditional society there's this unhealthy attitude that anything older is automatically better. So you created something new, a lot of people would say "who are you to change things that have stood the test of time, you think you're better than everyone who has come before you?!" To get around this problem, people would refuse to name their teacher. Again, withholding that info is not the same as lying about it (I learned it from actual person Y), which would be qi shi mie zu. Sometimes, if really pressed, they would say "I learned it from this traveling monk/daoist with special abilities while I was traveling in such and such remote/sacred mountain". Novices eat this up, because now it's all mystical. But insiders know that's code for "I came up with all of this".
4. The student's skill is not mature yet. In China, a lot of times students are not prohibited from going outside the group to test out his skill. The idea is, to really understand your skill, you need to try it out on as many people, with many types of opponents as possible. That said, before your skill has matured, your teacher usually say "don't tell people who your teacher is, you are not good enough to represent me yet", or "only let people know if you win".
5. The teacher is not famous. Fame in previous centuries is not like fame now - you can become well-known for publishing a book, have a popular blog, active on forums. Back then everyone knows who the best fighters are, just like today we know who the best basketball players are - they are the ones who fought other famous fighters, have good fight records. So famous fighter = good fighter. Life is not like martial art novels, where you can practice by yourself in some remote mountain, never touched hands with anyone, and just become invincible from day one. So if you're not well-known inside the martial art circle, it's safe to assume your real life fighting skills are not at the same level as people who are well-known.
So if one's teacher is not famous (he could still be good, even if not elite level), that's when you see people hiding that fact, or give you some mysterious explanation.
For that reason then, people like well-established school (sustained, excellent fighting reputation) is that there things are very transparent in this regard. They have no reason to hide any lineage information.
6. The teacher is infamous. Martial art people are very conservative when it comes to reputation. If your teacher's has bad personal reputation (say cooperating with foreign occupiers), even if he's a good fighter, that might be enough reason to hide it.
7. Special circumstances: sometimes the teacher has problems with powerful enemies and does not wish to broadcast his whereabouts. The most famous example is the first known Baji Quan master Wu Zhong. He was taught by a traveling couple with obviously made up names. Most people think it's because their real names were well-known to the Qing Dynasty government as loyalists to the previous dynasty.
As a disciple, the first duty is to fully inherit the teaching so one can 'guang da men hu' - bring honor and fame to the school, expanding it in the process. How can you bring honor to your father/teacher if you don't let people know who he is?