I don't know about 19th century Europeans but to 17th century Russians that was a serious issue because being formally a barbarian nation meant the Chinese wouldn't sign any treaties with you.

In 1618 the Russians tried to get received as ambassadors in Beijing but they were sent back because they arrived with no tribute. The Chinese would have gladly recognized them but only as a tributary state. Forty years later the Russians had settlements on the Amur river (current Russian-Chinese border) and the embassies got regular but still with no success (and with incessant border skirmishing, with the outnumbered Russians getting disarmed and convoyed to the North). It took a change of dynasty and a massive civil war (the Revolt of the Three Feudatories) to weaken China enough for a border treaty with the Russians to happen in 1689, recognizing everything to the North of the Amur river as the "eluosi state" (which still is the Chinese name for Russia).

Source: O.P.Okladnikov (editor) 1969. "История Сибири с древнейших времен до наших дней". Nauka: Leningrad.

Interesting question.

To answer bluntly, yes, Europeans were well aware that the Chinese, or at the very least, the Chinese Emperor believed they were barbarians. However, you should also note that the policies and beliefs of different Emperor's varied when relating to different countries. For instance, the Dutch and Portuguese maintained a permanent mission to China and the Jesuits were, by the 1700s, integrated in the imperial Chinese court. However, the general belief is most aptly shown when examining the Macartney Mission (British) to Qianlong's Court in the mid-1790s.

While this may seem a bit strange, I want to first begin by discussing the results of the mission, then moving to the mission itself.

The Result of the Mission

The Qianlong court was very clear about the British requests during the mission (more on that later). In Qianlong's Letter to King George III in 1792, the Emperor denies all British requests and refers to England as a barbarian nation. The letter is not very long, and I highly suggest reading it in full. Here are a few highlights:

First,

Hitherto, all European nations, including your own country's barbarian merchants, have carried on their trade with our Celestial Empire at Canton. Such has been the procedure for many years, although our Celestial Empire possesses all things in prolific abundance and lacks no product within its own borders. There was therefore no need to import the manufactures of outside barbarians in exchange for our own produce.

Second,

But your Ambassador has now put forward new requests which completely fail to recognize the Throne's principle to "treat strangers from afar with indulgence," and to exercise a pacifying control over barbarian tribes, the world over.

Third,

Consider, moreover, that England is not the only barbarian land which wishes to establish ... trade with our Empire: supposing that other nations were all to imitate your evil example and beseech me to present them each and all with a site for trading purposes, how could I possibly comply?

Fourth,

The distinction between Chinese and barbarian is most strict, and your Ambassador's request that barbarians shall be given full liberty to disseminate their religion is utterly unreasonable.

Fifth,

Do not say that you were not warned in due time! Tremblingly obey and show no negligence! A special mandate!

The fifth quote is especially telling: Emperor Qianlong refers to King George III as a barbarian subject.

By now, you should have some idea of what the British were requesting. Of course, the most important issue was trade. The British wanted to be free to introduce products into China. The East Indian Company were beginning to trade in opium (hint hint hint), but Maccartney himself had moral reservations preferring to try and introduce other goods.

1. A permanent outpost in modern day Hong Kong where the British could conduct trade out of. In addition, Hong Kong would be ruled under British, not Chinese, law;
2. An opening of other cities and areas to trade. At the time, the Chinese had put in place the Canton System, which limited European traders to Canton (Hong Kong). All foreign trade was to be conducted from Canton. The system was enforced rather effectively;
3. Establishment of reciprocal embassies. In other words, the British wanted to bring the Chinese into the Westphalian system with it's emphasis on the equality of nations. This directly struck at the heart of the Chinese worldview in which China was the "center of the world";
4. Allow Christian missionaries to freely work across China (note: while the Jesuits had been active since the 1500s, but were generally confined to the Imperial Court).

All British requests were denied and the mission returned to England.

Two Ceremonial Issues

To display how advanced European products were, the British brought a number of gifts to be presented to the Emperor (think things like clocks, telescopes, cannons). The idea was to show that British inventions had utility. Qianlong accepted the gifts, but referred to them as "tribute." So, rather than being a transfer of goods in an equal manner, the exchange demonstrated that Qianlong believed the British to already be a subject kingdom, while the British approached on equal terms.

There was also an extensive debate over how the British would approach the Emperor (and receive his messages). The custom at the time was to kowtow to any imperial messenger and to the emperor himself. The British were not fans of this. Macarteny was willing to accept the ritual, as long as both the Qianlong Emperor and King George III were presented as equals. In other words, the mission would kowtow to the Emperor if members of the imperial court would kowtow to a picture of King George III. The Chinese said no (for reasons that should be relatively clear by now.) It was eventually decided that the British mission would genuflect in front of the Emperor, but not without significant debate.

The Tumultuous 19th Century

Speaking broadly, the Chinese belief in superiority was shattered by the Opium Wars and the Sino-Japanese War. In the aftermath of the Opium Wars, the Chinese Emperor was forced to recognize European Countries on equal terms and open the country to freeish trade. In essence, the Opium Wars brought the Chinese Empire into the Westphalian System. They could no longer claim they were the most powerful country in the world. In the aftermath of the Sino-Japanese War, they could no longer claim they were the most powerful country in Asia, let alone the world. Overall, these three wars shattered the official Chinese worldview, undermined the strength of the Emperor and set the stage for the 1911 Revolution.