You're addressing this question by two different standards. Instead of Romance languages you should say European languages, in which case then the answer is clear that not all European language share a common known ancestor. Or you should likewise narrow down to a specific family of languages in Asia like you have in Europe.
Sinitic languages (a.k.a. Chinese languages) do share a common ancestor amongst themselves. This includes Mandarin, Cantonese, Hakka, Wu (Shanghainese), Min (Teochew, Taiwanese et al), Xiang (Changsha), Gan and a few others.
It has often been said, most notably and early on by Chao Yuen-ren, that Sinitic varieties really are a good analogy to Romance in their geographic spread and internal diversity.
Quoting from Jerry Norman's Chinese (1988):
The Chinese dialectal complex is in many ways analogous to the Romance language family in Europe: both have their roots in large-scale imperial expansion that took place in the centuries just preceding and just following the birth of Christ, ... in both instances the imperial language was carried by armies and settlers to areas previously occupied by speakers of different languages; in the course of their development both were affected by the "substratum languages" in both cases the newly developing vernaculars existed alongside an antiquated written language and were profoundly influences by it. …… There us probably as much difference between the dialects of Peking and Chaozhou as there is between Italian and French; the Hainan Min dialects are as different from the Xi'an dialect as Spanish is from Rumanian.
The Sinitic languages are then related further back in time to the Tibetoburman Languages like Tibetan languages, Burmese, Qiangic, Ngwi (Loloish/Yi) and many others.
You can take Sinitic to be roughly equivalent to Romance, and you can take Sinotibetan as roughly equivalent to Indo-European.
However there are a number of languages in East Asia which do not fit into Sinotibetan. Koreanic languages (modern day Korean dialects being the main surviving branch) are isolated. Japonic (Japanese, Ryūkyū) are also isolated. Austronesian languages are spoken throughout places like Taiwan and Southeast Asia. The aboriginal Taiwanese languages are related to what's spoken in Hawai'i and Madagasgar, but not part of the Sinotibetan family or any other in the region.
Then you have Hmong which is part of the Hmong-Mien aka Miao-Yao family. This sometimes gets classified in with Sinotibetan (at least in Chinese sources) but is not generally considered part of Sinotibetan by modern standards. Tai-Kadai is another separate family, which includes Thai, Lao (spoken in Laos) and Zhuang (spoken in China). These are also a separate family.
And this is all just in East Asia. I'm not at all addressing Central Asia, South Asia, Siberia or other places which only further adds to the number of language families.
To answer your question then, as you worded it, yes, some Asian languages share a common ancestor like Romance languages do from Latin. However others do not, just like not all European languages share a common ancestor in Latin.
Asia is massive and linguistically complex, but then so is Europe. So it would be better to make sure your thinking of comparable scales of geography in order to better understand what's happening.
Sources
Chao, Yuen Ren. "Languages and Dialects in China" In Aspects of Chinese Sociolinguistics, ed. Anwar S. Dil. Stanford: Stanford University Press, 1976.
Norman, Jerry. "Chinese". Cambridge Language Surveys. Cambridge University Press 1988. p 187
edit: oops. page numbers.
Let me start by saying that the languages of Asian belong to many different language families. In some countries, such as India, Pakistan and Iran the majority speak a language that belongs to the Indo-European language family. The Indo-European language family is a large collection of languages in Europe, Asia, and America and Oceania through colonialism, that share a common ancestor (Called proto-Indo-European, who according to the standard theory(Kurgan hypothesis), was spoken by people who lived somewhere close to the Black Sea), with varying closeness of ancestry, for example Persian and French share proto-Indo European as their common ancestor, but French and Spanish share a more recent ancestor in Latin, but Latin also has proto-Indo European as its common ancestor with Persian so they are all related if you go far enough back. (Think of it as a tree where Proto-Indo European is the stem, Latin and Persian as branches, and French and Spanish smaller branches from the Latin Branch) So, the Asian languages that belong to the Indo-European language family do share a common ancestor. There are also other language families in Asia, but i'm less familiar with them, but some examples I can think of right now (if you include the Middle east as Asia) are Hebrew and Arabic which both belong to the Afroasiatic language family, and thus share a common ancestor. Turkish also shares a common ancestor with the other Turkic languages (such as Kazakh, Uzbeck and Azeri) in Central Asia.