Sure, but it's family based. There are things like Old/Middle Chinese. Middle Chinese became Cantonese, Mandarin, Hakka, Wu, Xiang, Gan, Ping, Jin... and Old Chinese became those plus Min, Ba-Shu, Jiangdong. There are plenty of cognates there. Korean and Japanese borrowed a lot from various Chinese varieties, and so through those also many cognates. Thai, Lao, other Tai varieties, Khmer, Burmese all have a lot of Pali loans as well, so through those also then many cognates, and of course Tai lent a lot of words into Tibeto-Burman varieties. There are also many cognates between Tibeto-Burman varieties and Chinese, since they are within a single family (Sino-Tibetan, as it's more traditionally known). So words like "eye" may be mit or mik or muk or mu, all cognates.

Then also you have Classical Chinese that served as a sort of Latin among non-Chinese-speaking groups. It was a written language used for official communications, even thought it wasn't what was spoken.

I do historical linguistics in Tibeto-Burman, but recently was Sinitic-focused, and often I see what appear to be obvious cognates. One of my current areas of study is cross-family loans, as well, which involves looking at the kind of lexical items which we can say with near certainty were borrowed, and which are almost certainly older words that are shared within a single family. I work mostly in Sino-Tibeto-Burman, but also end up dealing a good bit with Tai languages as well as some Indo-Aryan, and Indo-Aryan languages have a lot of borrowing from languages like Arabic as well. These are mostly easy to work out, and in most cases cognates are also easy to work out, but it's not always so clear.

Are there any asian countries that have a similar situation that Spain has with the Basque in that they have totally different languages and cultures despite living together for centuries.

Languages have families. Think of family trees. So Spanish and Basque are in different families, but Spanish and Italian are in the same family. Same thing happens all over the globe.

If you have another cool fact or a question and answer I should've put in my post. Leave if in the comments :)

Your post actually is mostly only asking one question, to which the answer is "yes", but if you have more specific things you're interested in, I'd be happy to expand on any relevant topic.

Your question is incredibly vast, and runs into the problem of what "Asian" is. Latin and Greek are both indo-European languages (Ancient Greek, by the way, only really has one modern descendant... namely Greek.) The most impressive example of an attested Indo-European "mother language" is easily Sanskrit, which spawned Gujarati, Hindi, Bengali, most likely all the Indo-European languages of the Indian subcontinent. The related Iranian languages have a similarly old language that has no known modern descendants, viz., Avestan. The most widely accepted theories suggest the Proto-Indo-Europeans were inhabitants of the steppe of Eastern Ukraine and Southern Russia, north of the Caucasus and the Black and Caspian seas, at least 4500 years ago.

As an addendum, the Indian subcontinent has a large family of unrelated languages as well - the Dravidic languages, such as Tamil. These are pretty comparable to your example of Basque.

A great book on the (pre-)history of Indo-European languages is The Horse, The Wheel, and Language.

If there's anything you'd like elaboration on, let me know!