Well, for example, in the Tale of Genji, which is from 11th century Japan, long before European influence or even European trade, pale, ephemeral white skinned women were described as beauties. Compare this with contemporary descriptions of common folk or the ugly, who are dark skinned.

Even longer ago, during the Nara Period (8th century Japan), court ladies would use cosmetics (techniques of which were imported from China, where the white skin craze was well underway) made from rice powder, lead, and even exotic ones made of bird droppings. All in the pursuit of white skin, especially the face.

Of course, there were other beauty traits besides white skin, such as straight, long, and luminous black hair, as well as black teeth! This black teeth fashion was one of the hallmarks of beauty that did not survive contact with the West, as Japan was highly self critical of anything that might embarrass them on the global stage when they were trying to become a world power. This is also the same time that Japan, unfortunately, rapidly embraced the racial sciences and were proud to announce to the Western peoples that they, too, were white (or at least Whiter than other Asian races).

The easiest frame to see the white skin beauty in is probably the world renown geisha, which definitely predated Western influences and entry to the country.

During the Ming Dynasty, Chinese women with great white skin were often referred to as 'jade white', and of course, there were many 'products' that enterprising young ladies could buy to help them achieve such great skin. Everything from creams, powders, special exotic medicines, even massages, and of course, food. To this day, Three White Soup (三白湯) is still sold as an effective, natural way to turn your skin whiter, healthier, and more robust. Traditionally it is made with white peony root, white poria, and atractylodes.

Many of the various regions in East and South East Asia had their values of beauty heavily influenced by China, because of the heavy connection between skin tone and class and general cultural diffusion by the most culturally dominant force in the region. This class distinction was not only used to differentiate within one kingdom, but outside of it as well. 'Lesser' kingdoms were often described as darker skinned, among other things.

As for today, while white skin is still the ideal for many places in Asia, including China, Korea, Japan, Vietnam, Philippines, etc., very few people would say they are trying to emulate white caucasian people from the west. They are trying to emulate the light skin Asian beauties. People in China may admire a white Hollywood star's white skin but this is completely unlinked from American and European concepts of a 'white race' and other race based social constructs.

As a side note, white people are not considered universally good looking in Asia, as disappointing that may be for some prospective English teachers.

The skin tones beauty standard in Asia has existed long before the introduction of Western beauty standards. Dark skin tones have long been associated with low class, pedestrian, peasantry, and 'dirty-ness'. Again, these standards were not linked with the concept of race (that is, the system where people were separated primarily by skin color).

If you have any questions, feel free to ask!

Here's one piece of independent documentation from the Tang Dynasty: Yan Liben's painting Emperor Taizong Receives the Tibetan Ambassador.

Now, this might be a slightly later copy of the original, and it's from a time when there had been some kind of contact with Romans and Jewish merchants... but nothing like what happened 1,000 years later.

The women surrounding the emperor are all fairly pale compared to the emperor himself, as well as in comparison to the Tibetans to the left.

(The painting is a little... politically rife, as you might imagine. It's part of a larger scroll. But the main thing here is that the feminine ladies around the emperor are paaaale.)

There are similarly pale figures among Zhou Fang's court ladies paintings, but there's less of a masculine/feminine contrast there, and Zhou was painting about a century and a half later.

One thing I'm a little curious about is how much lighter-skinned your average Roman traveler would have been in the 7th or 8th century. Seems like you'd get plenty of sun, especially if you're sailing most of the way there.

/u/AsiaExpert gave a great description for East and Southeast Asia, especially Japan. I'd like to weigh in add a bit more on China. The earliest reference to "white" skin is in the Classic of Poetry, a collection of ~300 poems compiled between 2600 - 3000 years ago. One of these poems is dedicated to a daughter of the Marquis of Qi, who is getting married to the Marquis of Wei. A translation by William Jennings reads the following:

Classic of Poetry - Book V - The Odes of Wei - Poem 3

An Epithalamium

Stately her person - tall and fair,
Clad in her robes embroidered and plain.
Child of the lord of Ts'i,
Bride of the lord of Wei,
Sister of the one who is Ts'i's next heir,
Sister of wife of the lord of Hing,
Same of the lord of the T'an domain!

Fingers, as softest buds that grow!
Skin, as an unguent, firm and white!
Neck, as the tree-worm's breed!
Teeth, as the gourd's white seed!
Mantis' front, and the silk-moth brow!
Dimples playing in witching smile;
Beautiful eyes, so dark, so bright!

Stately in person - proud and free,
Halts she awhile in the lands close by,
Her four male steeds, full-fed,
Bright in their trappings red;
Screened by her plumes, then to court comes she.
Haste, ye councillors, quick depart;
Do not your chieftain's patience try!

Wide and deep is the river, the Ho,
Northwards flowing, majestic, grand.
Broad nets splash for a haul!
Sturgeons are in, big and small!

It should be pretty clear that these words are meant as a compliment to the bride. So you see, the idea that lighter skin tone associated with beauty is a very ancient one in the East Asia. Certainly has nothing to do with European colonialism and racism. Powder to whiten face was referenced in A New Account of the Tales of the World ~1600 years ago. So, these products would have a long history as well.

Edit: William Jennings' work is The Shi king, the old poetry classic of the Chinese. A close metrical translation, with annotations by William Jennings
It's out of copyright and you can find it here.

Is this a uniquely Asian phenomenon? Did it exist in Europe too before industrialisation? Did it exist in other cultures too?

Did this and colonial attitudes/racism begin to intersect at some point?