So, with this Lore of the awoken, would their old Cultural background be more based on Asian on European descent?
The Awoken didn’t retain any specific cultural background. The original crew of the Yang Liwei was pretty mixed and the only piece of pre-Distributary heritage to survive was the Amrita charter with its foundational principles of peace, equality, and responsibility. Though tech/science survived in the Shipspire, nothing else did. All Awoken memories of pre-Distributary life, plus the records of any previous civilizations or cultures, were wiped out in the transition - hence the name “Distributary,” meaning a river that diverges and doesn’t return to a larger waterway. The Sanguists consider the loss of history to be a boon and a necessary part of the covenant that granted them their world: “we yielded our claim to our history. We abandoned what came before, but in doing so, we cast off all our debts.” The Eccaleists, on the other hand, believe the Awoken still had a responsibility to their origins that they would one day be called to fulfill. Neither side particularly mourns the loss of culture, though - likely because they didn’t know what culture they had to lose - but when the Awoken return to our solar system the faction that heads to Earth does so both to help their ancestors and to rediscover their own roots.
Speaking purely in game design terms, I think the Awoken aesthetic is carefully crafted to evoke beauty without hewing to any specific culture. It has an Asian feel of seeking to perfect technique and craft, of honing a skill or design ideology down to its purest forms over thousands of years, but much of the actual bits and bobs are Western or generic in origin. The buildings of the Dreaming City evoke European cathedrals in their pointed arches and stained-glass, especially the rose windows, and the Gardens of Esila have several planned landscapes that resemble formal European gardens with benches, planters, etc. They also have a lot of more modern pavilions, round open structures supported by plain columns and Roman arches. But the Awoken also really like obelisks and pylons, which Europeans didn’t go for (as far as I know), and they use a lot of purple, which I don’t think anyone went for because you don’t get to use a lot of purple unless you carve your city out of amethyst, so. The cleft gate motif - the recurring Awoken structure of two tall semicircular pieces with no crossbar standing on either side of an entrance, symbolizing the separation between past and present - has more in common with Asian latticework or shrine gates than European wrought-iron. Awoken also tend to use natural features like uncarved crystals or carefully-grown trees as visual centerpieces rather than manmade structures.