Not exactly in those terms, no - from the perspective of the Byzantine Empire, nothing really significant happened in 395 that would be worth celebrating in 1395. The Byzantine Empire was a continuation of the Roman Empire, which we now traditionally date to 27 BC when Augustus adopted the title emperor, but for them this was just a continuation of the Republic, which was just a continuation of the monarchy, going all the way back to the founding of Rome itself on April 21, 753 BC. So 753 BC was the only date they would recognize as the start of the Empire. And indeed, they did celebrate the 1000th anniversary of the founding of Rome on April 21, 248! (There was a question about this recently that I didn't have a chance to answer - but yes, emperor Philip the Arab hosted lavish celebrations in Rome in 248.)

In 395 the empire was split into two halves with one emperor ruling in each half, but that was nothing new, the empire had been ruled jointly before (and about 100 years before that, it was ruled by *four* emperors at a time). We tend to look at 395 as the start of the Byzantine Empire today, but that’s a modern convenience, that year wouldn’t have meant much to them.

I suppose they could have celebrated the 2000th anniversary of Rome in 1248, but Constantinople was controlled by western crusaders at the time, and the various Byzantine successor states were a bit preoccupied trying to get it back. They did get Constantinople back in 1261, but the Empire never really recovered. By 1395 Constantinople was basically completely surrounded by the Ottomans. Constantinople itself was under siege in 1395. Emperor Manuel II spent a lot of time begging for help in Western Europe. His two sons succeeded him and they were the last emperors before the Ottoman conquest. Nobody could have been in a celebratory mood in 1395, even if they had recognized 395 as the founding of the empire.

In any case, by then the Byzantines no longer used the ancient Roman “ab urbe condita” system. The empire had been thoroughly Christianized and Hellenized so the founding of Rome didn’t really matter anymore. They used the “Anno Mundi” dating system instead, which calculated the creation of the universe to 5509 years before the birth of Jesus. The idea was that "a thousand years are but a day in God’s sight” (as it says in the Bible), so every thousand years corresponded to a day of the week in the creation story in Genesis. They were, therefore, very concerned with what would happen in the year 6000 AM, or 492 AD, and for the time period you’re asking about, they were also very concerned about what would happen in 7000 AM, or 1492 AD. Presumably something apocalyptic would happen then, or at some point before the end of that millennium - and indeed it did, when Constantinople finally fell to the Ottomans in 1453.

See Paul Magdalino's chapter “The Year 1000 in Byzantium”, in Byzantium in the Year 1000 (a collection edited by Magdalino), for a discussion about what kind of dates were important to the Byzantines.

a follow-up question: I've heard that they were calling themselves Romans and the name of Byzantine is something the historians made up much after the fall of Byzantine empire. Is it true?

Also, did the way the locals and nobles referred to the Byzantine empire change over the centuries? If yes, how?

The Byzantines were acutely aware of the depth of their history. But since they always understood their state as a continuation of the ancient Roman Empire, they never celebrated any kind of anniversary for the establishment of a "Byzantine" state. The closest they came were the annual ceremonies that marked the foundation of Constantinople.

To the very end, famously, the Byzantines called themselves Romaioi (Romans), and neither the emperors nor their historians ever forgot that the lineage of the Empire stretched back to antiquity. As I've discussed in a previous answer, only a few scholars knew much about the more distant aspects of that legacy (like the civil wars of the Late Republic). But consciousness of a deep, if imperfectly understood, past was enduring. Perhaps the most poignant witness is a letter in which the Emperor Manuel II (r. 1391-1425), forced to campaign alongside his Ottoman overlord, wistfully describes the Roman ruins along his route:

"The plain where we are staying certainly had some name when it was fortunate enough to be inhabited and ruled by [us] Romans. But now, when I ask what it was....nobody knows....You have heard of the city founded by Pompey: beautiful, marvelous, and extensive - or rather, that it how it once was, for now you can barely make out its ruins...indeed, this city and its magnificent remains offer no less evidence why the Romans bestowed on its founder the surname of "the Great" than the many victories that amply justified the title" (Letter 16)

It is clear that this Byzantine emperor was deeply aware of his Empire's history.

Moving on the question of anniversaries. The Romans famously celebrated the 1000th anniversary of their city in 247 CE with a great fanfare (we are told that 32 elephants and more than 1000 gladiators were involved). The Byzantines did not mark the anniversary of Rome's foundation (despite the fact that Rome itself was part of their Empire for several centuries). They did, however, commemorate the foundation of Constantinople.

Constantine had dedicated his new Rome on the Bosporus on May 11, 330 CE. The rituals of dedication (which lasted 40 days) culminated in a grand ceremony in the Hippodrome, during which a gilded statue of Constantine, standing in a golden chariot, was escorted around the track by a company of imperial guardsmen bearing lighted candles. This ritual was repeated every May 11th for centuries: Constantine's statue in its chariot would make a full lap of the Hippodrome, flanked by the guardsmen and their tapers. As the statue passed, the current emperor and his courtiers would prostrate themselves. This tradition went on into at least the sixth century (though eventually the habit of prostration, with its implications of pagan imperial cult, was dropped).

I am not aware of any rituals staged in 1330 to mark the millennium of the city's foundation. By then, after all, the Byzantines had many other things to worry about...

First, it helps to understand that both the terms 'Byzantine' and 'Empire' are both modern terms used to describe something that thought of itself differently.

Enough has been said about the invention of the word 'Byzantine' by Western historians and its use to interpret history, but perhaps not enough has been said of its use to interpret history as they saw it. If we explore a little here, we can see a lot of historical understanding and modern identity are built on the belief that there exists a continuity from Ancient Greece, to the Roman Republic and Empire, to the medieval Kingdoms, and to the Western Civilization and the concept of it.

It is important to understand this belief in continuity to understand the world view of those writing history and their belief in being inheritors of, in this case, Romanity. This colors all their understanding of their place in the world and bleeds into anything they might write. Which is why I feel it important to look to the sources, as far as possible, to get a better understanding of these peoples and how they understood themselves.

Answering this question is tricky in this sub because most authorities on this subject reinforce, to some degree, tainted ideas. I hope that the moderators allow this comment to remain because I believe it's important to challenge historical understandings, when they exist because that is how they were taught.

As an example, I'd like to point to Anna Komnene's work, the Alexiad. In the opening line of the first book, Anna refers to the βασιλείᾳ Ῥωμαίων. This translates to the Kingdom of the Romans. In the English translation of the Alexiad, the translator, Elizabeth A. S. Dawes, uses the Roman Empire to describe the same entity. Why did Dawes translate it like this? I can't say for sure, but one can make an educated guess and say that that was the accepted term. Kingdom of the Romans sounds very different and unusual to our modern ears, and yet this is how the people themselves understood their State. This is a small example that illustrates how subtle changes, even innocent, can significantly alter the image we see. I like to think of it as 'coloring' the 'image'.

The Wikipedia entry you refer to neatly defines the 'Empire' with start and end dates, but this, like the above, is a kind of revisionism. It states:

Theodosius I (379–395) was the last Emperor to rule both the Eastern and Western halves of the Empire.

and goes on to define East and West as separate States with the end of the reign of Theodosius as being the start date of the 'Empire'. The people at the time didn't view it this way. When Justinian went on to conquer lost territories he did so to recover Roman lands for the Roman State that existed long before Justinian or Theodosius or even Constantine, who some also like to use as a start date. Thus, the use of the year 395 to date the beginning of a new State entity is incorrect.

If we do away with artificial time stamps, we're free to better understand how the people viewed their place in history. We know that the medieval Romans paid a lot of attention to time and produced several known chronographies that detailed events throughout history and tried to date the history of the world. In fact, the creation date was used as the official calendar of the State so we can surmise that they were well aware of the age of their State.

So your question:

is there any evidence the Byzantine Empire aware it had lasted this long and if so did it do anything to mark the occasion?

we can say that for the first part, yes, we have evidence of this. As to what date they considered to be the beginning of their State, it certainly wasn't 395 as the article states, but was thought of more as a direct continuity from the founding of Rome.

As to the second part, I don't know. I can't say that I've come across any event like you describe before the above poster's answer.

I hope I've answered your question.

sidenote: I used the term 'State' a lot in this answer as I feel that 'Empire' is technically incorrect. I understand that 'Kingdom' would probably be too confusing so I think a better option would be to use 'Basileon' with 'Basileus' in the place of 'Emperor'. I'm not sure what other people's thoughts on this are, but I'd like to hear other people's opinions.