It’s a very interesting question. My take is that the modern view of Qing is quite China-centric, in the sense that Qing is viewed as a Classical Chinese dynasty instead of a multinational empire. The Manchus did use the name of Qing in the capacity of a traditional Chinese 國號 or official state name, which is an important formality for ruling over Confucian China regardless of whether the state viewed itself as Chinese. Adopting a Chinese state name was more or less a standard practice for any foreign state that consistently ruled at least a significant part of ancient China. AFAIK the only exemption was the Tibetan Empire which did not attempt to sinisize its government when it temporarily controlled vast portions of China proper during mid Tang Dynasty.

What complicates the matter is that there are ongoing debates about the usage and meaning of the name “Qing” outside the Chinese language, which reflects on the Manchurian identity.

The Khitans, for example, adopted the Chinese state name of Liao and used it extensively in their Chinese inscriptions (e.g. in Chinese bios on tombstones of Khitan nobles), but only referred to themselves as “Khitan” in the Khitan language. The Mongols, on the other hand, incorporated their Chinese state name of Yuan into the official Mongolian name of their empire. Both Liao and Yuan are names with strictly Chinese origins. In the former case, Liao was merely a tool through which Khitans communicate with their Chinese subjects. In the latter case, Yuan was a carefully chosen symbol of Mongolians transforming into a Chinese empire.

Qing is unique that although the name does have Chinese philosophical implications, it’s also a clever Mongolian wordplay (“Daicing,” meaning “Greater Qing” in Chinese, also means “warrior” in Mongolian and Manchurian). The name “Greater Qing” or “Daicing” was well-known among Mongolians and Manchurians during the Qing dynasty, and the connection to their own languages was definitely appreciated. But on the other hand, “Greater Qing” or “Daicing” was not the preferred nomenclature in contemporary Manchurian or Mongolian languages, as Qing rulers preferred using “China” or “Middle Kingdom” in non-Chinese documents to refer to the empire as a whole. This was a significant departure from previous norms. The Manchus were not trying to play around a well defined cultural line separating the Chinese and the others like the Khitans and the Mongolians did. Instead, they were trying to define that line differently when dealing with different groups of people to suit their needs. They were selling the idea of a classic Chinese dynasty with a Chinese-Barbarian dichotomy world view to the Han Chinese, and a concept of a multi-ethnic empire with a multicultural Chinese national identity encompassing the Manchus, Mongolians, Han Chinese, Kazakhs etc to the Mongolians. The Qing name plays a part in both of these visions they tried to promote.

TL;DR: The Manchus referred to their empire as “China” or “Middle Kingdom” in non-Chinese texts, and as “Qing” in Chinese texts out of necessity. We call them “Qing” because historians of the 20th century largely viewed the empire in the context of a Chinese dynasty.