Short answer: Yes...Maybe?

Long answer:

So before getting into the question itself, we should define the question. The easy one is the time period. Due to Nobunaga's influence on his successors, we'll just limit the time period to before Nobunaga's death. The next one is "reunification". The term used, 統一 (tōitsu in Japanese) does mean "unification", but pre-modern Japanese liked to crib terms from China, so it's important to put them in the Japanese context. Officially Japan was not "divided". That might sound weird with the all-out war all over the place, but no one, challenged the authority of the Emperor and his court. The emperor still reigned over the empire. This is easy to see as Japan still used a unified reign year, unlike Chinese dynastic conflicts (or actually the Nanbokuchō too). The conflict was over who ruled, and in the provinces, who ruled what. There's no evidence that Nobunaga was going to declare himself Emperor. There's also no evidence that Nobunaga was aiming to re-institute direct court rule over the provinces using court-appointed administrators rather than hereditary local lords. So, when the dust settled, in a sense Japan would have been (and indeed as it turned out, was) just as united/divided as before. Now all of this doesn't change the fact that Ashikaga Bakufu's authority had been heavily damaged (and as it turns out, beyond repair), and that the Bakufu's power was not strong enough to keep the peace and enforce jurisdictional borders. But, given the above, the aim should be thought of as the (re)creation of a central ruling authority capable of ordering the lords around and keeping a general peace. Though I'm going to keep using the term "unification", rather than "unification" in the Chinese sense, "pacification" is probably closer in meaning. This might seem a bit pedantic, but it actually throws a curve ball in our answer.

Now on to the question. It's pretty clear by everyone's action that everyone was interested in local control and fought for it. So how do we know that Nobunaga was interested in "unification"?

The clearest answer is because he worked towards it, moving his power base from the Nōbi plain into the Kinai, closer to Kyōto, and sending armies in all direction to force lords to submit, and just before his death even sending orders (and receiving a couple of positive answers) from lords all over Japan to submit to the new ruling authority. But this is kind of arguing from hindsight. Nobunaga was the first one to have the power of doing this, and he was the first one to do this. So, at least just before his death, it was pretty clear he had unification as a clear, immediate goal. No one else (before Nobunaga) did, so no one else could be proven to have had unification as a clear, immediate goal. But also, no one else was in a position to do so before Nobunaga. And Nobunaga, like everyone else, was once upon a time also not in a position to do so.

So how do we know that other lords didn't have the "intention" or "goal" of eventually unifying and constructing a new ruling authority. To answer this (or get more confused), let's take a look at how we thought we know Nobunaga had eventual unification as a goal. First, the famous "Tenka Fubu" seal. It's usually thought (and translated as) something along the lines of "unifying the realm through military force". However, just in the past few years, a couple of Japanese scholars has challenged this idea. The reason is quite simple. If that is what "Tenka Fubu" meant, and that's how people would interpret it, then every communication stamped with the seal to someone who was not a subordinate would effectively be either an ultimatum to submit or a declaration of war. Yet the seal was used for friendly communication, and even communication requesting aid and an alliance. It's a bit hard to imagine Nobunaga would say "Hey, can you please help me with our common enemy? Oh and by the way, I'll eventually conquer you if you don't submit to my authority."

Breaking the phrase down, 天下 "Tenka" means all under heaven. So there's no question that Nobunaga had design for the entire of Japan. The challenge is what 布武 "Fubu" means. Though it is made up of the characters for "spread" and "military", a couple of recent scholars argue we should be looking for what those reference in traditional Chinese text, as was usually the case for Japanese back then. I'm just going to wait for this dust to settle on what the meaning for "Fubu" is, but accept that it probably wasn't something that could be taken as an ultimatum to submit or declaration of war. Nobunaga was not declaring his intention for unification in the Chinese sense. Perhaps something weaker, like "restore peace to the realm" (which would be more in line with our stated definition of unification).

Even that, of course, doesn't change the fact that Nobunaga had unification as an eventual goal. But that's not the question here. It is whether other lords had the same goal. I am not aware of any surviving examples of other lords who put that into writing, or a seal like Nobunaga did. But let's examine the background of when Nobunaga started using that seal. Nobunaga started using the seal late in 1567, after completing the conquest of Mino. In 1566, Ashikaga Yoshiaki had escaped Kyōto after his brother and Shōgun Ashikaga Yoshiteru was assaulted and killed in a coup. Yoshiaki sent out calls for lords to drive out the usurpers and support him as the new Shōgun. Nobunaga answered the call, or intended to do so, but fighting in Mino blocked his way, forcing Yoshiaki to look elsewhere. Late in 1567, Nobunaga finished his conquest of Mino and began using the seal. In 1568, Yoshiaki once again called on Nobunaga for help, and this time, Nobunaga supported Yoshiaki, brought an army into Kyōto, kicking out enemies along the way, and welcomed the new Shōgun. For the next couple of years, Nobunaga was nominally Yoshiaki's subordinate, working to spread and restore the rule of the Muromachi Bakufu.

Yoshiaki and Nobunaga eventually turned on each other. However, it's important to see that whatever "Tenka Fubu" actually meant, it was understood as something that would not exclude Nobunaga restoring the rule of the Muromachi Bakufu, or else Yoshiaki would never have came to Nobunaga in the first place. Now is it possible that Nobunaga had a different, personal interpretation for "Tenka Fubu" compared to the accepted definition in 1568, and he always planned to have an eventual confrontation with Yoshiaki? Well, sure. It is possible. But it's not provable (unless someone finds Nobunaga's diary). And if that were the case, Nobunaga sure didn't prepare for that confrontation very well, being caught on the back foot when Yoshiaki arrayed an alliance against him. So let us assume, and I think justifiably, that Nobunaga's personal definition for "Tenka Fubu", at least in 1568, also did not exclude restoring the rule of the Muromachi Bakufu over Japan. So, on adoption of the Tenka Fubu seal, Nobunaga's stated goals should be considered as "I intend to bring peace to the realm under the name of the Muromachi Bakufu (Ashikaga Yoshiaki)".

This, of course, brings us to the dying Muromachi Bakufu. It was dying, but it was not without influence. It still commanded respects, and sometimes (more often than usually thought), it's orders were even followed. Nobunaga, after all, followed Yoshiaki's call to arms. If the Muromachi Bakufu had no authority, no one would bother trying to install their own (puppet?) Shōgun. But ever since the end of the Ōnin War, lord after lord brought their armies to Kyōto to install a new Shōgun, only to be kicked by another lord out after a while. If Nobunaga's original stated goal was to "restore peace to the realm under the Muromachi Bakufu", then should we not consider these previous lords that did the same? Even if the Hosokawa, Ōuchi, Miyoshi, Rokkaku, Asakura, and others didn't specifically state that their intention was unification of Japan, I think the argument could be made that they intended to rule in the name of the Muromachi Bakufu simply by the fact that they actively made a bid to control the Muromachi Bakufu. And since the Muromachi Bakufu nominally ruled Japan, I don't believe it is a stretch to think these lords prior to Nobunaga also thought of eventually, if everything went all according to plan, to "restore peace to the realm under the Muromachi Bakufu", or something similar.

So why is the answer not a "no"? First, because this can't be concretely proven. As far as I am aware from current scholarship, none of these lords displayed a clear sign for "unification" or some design for the whole of Japan prior to Nobunaga. Nor were they able to use the Muromachi Bakufu for much more than influence in the Kinai, and very weak at that considering how soon they were kicked out. But for the latter is important to remember that the same could be said about Nobunaga at one point, and only after surviving the alliance Yoshiaki arrayed against him and dispensing with the Muromachi Bakufu did it become clear that "unification" and the creation of a new ruling power would be a clear, concrete, and kind of immediate goal. So in that sense Nobunaga was still unusual.

Second, it's important to remember that the vast majority of lords were actually too physically far from Kyōto to do anything more than visit (if at all). So even if all lords who made a bid for control of the Muromachi Bakufu had "unification" in the back of their minds, that would still make Nobunaga member of a very exclusive group of lords who did. So in that sense, even if not the only one, Nobunaga would still be "not usual".