Advaita Vedanta of Shankara of Hindu philosophy is monistic. It postulates that the world is utterly manifestation of Brahman, and the sense of separateness of the Souls from Brahman is an illusion.
On the other hand, Hegel postulates that the world and the Souls are but manifestation of the Absolute.
Could we consider that both Hegel's Absolute Idealism and Shankara's Advaita Vedanta are identical?
Hegel's system is essentially a continuation of Spinoza's system, with a teleological twist. In Hegel's system, there are tensions that are resolved and subject's evolving to an ultimate purpose. In Hegel's system subjects positively advance throughout history, in order to achieve some telos, the realisation of Spirit. Hegel accomplishes this by encouraging communitarianism and improving the society's arrangement (i.e. to achieve the nation's spirit).
It is incompatible with Vedanta because there, historical evolution by such progress is considered negative affirmation with the material world (the illusion). Brahman is already a perfect ultimate reality, and by our profuse karmic actions, we are fostering the larger dualism, we spur the clinging to our bodies, passions and goals, thus dissociating from the One. The cure is detachment from these:
One who performs his duty without attachment, surrendering the results unto the Supreme Lord, is unaffected by sinful action, as the lotus is untouched by muddy water.
— Bhagavad Gita
One who sees inaction in action, and action in inaction, is intelligent among men.
― Bhagavad Gita
Hindu sages are frequently ascetics who deprive themselves of food or other life pleasures in order to release themselves from the illusion of changing world (Maya) and reunite with the whole. Through meditative absorption that extinguishes the attachment, they become one again and embrace Moksha- the liberation from the illusion (also the illusion of history).
When a man dwells on the pleasure of sense, attraction for them arises in him. From attraction arises desire, the lust of possession, and this leads to passion, to anger.
From passion comes confusion of mind, then loss of remembrance, the forgetting of duty. From this loss comes the ruin of reason, and the ruin of reason leads man to destruction.
― Bhagavad Gita
Hegel's system does not have any such negative concept of karma, action, or duality, and doesn't consider the world an illusion (because history is a real manifestation of Spirit). Rather, it embraces material affirmation, progress and changes. It goes through history to achieve the restructuring of human's society via ongoing positive process of perfection, in order to best represent Spirit.
With Hegel off the table, the direct inspiration and influence in Western philosophy would be Schopenhauer's idealism since he based his philosophy on large portions of Advaita Vedanta (and Yogacara).
In Schopenhauer's monist system, the Will (thing-in-itself) is akin to Brahman, and the liberation is achieved similarly; through either antinatalism, to reduce the suffering, or inaction through meditative absorption, i.e. immersive contact with the beauty. This is akin to Moksha, and Schopenhauer also explicitly embraces the Eastern notion of non-duality.
Why say they are 'identical'? That's like lumping Akhenatan worship and Sikhism together, as 'identically' monotheist.
I think there is a deep parallel between the motivations of Shankaracharya and Plato.
Plato with his Forms, and the monism of Shankaracharya, are creating a framework which can relate cultural & religious diversity, into a single structure. While not consciously motivated by political aims, I'd suggest this methodology fits a kind of 'spiritual entrepreneurialism', where it gained attention & support for providing a basis for cultural unity, without requiring homogenisation. Essentially, a means for more tribes to collaborate more effectively.
The Islamic Ummayyad's were able to go from being a desert tribe confederation to ruling 29% of the world's population in less than a century of becoming monotheistic, discussed here: Was the ancient Jewish concept of God utterly unique?
Hegel had a different motivation. In a post-renaissance world, he was providing justification for the myth of progress, and 'manifest destiny' of colonialism. That's why he was enormously intensely popular in his own lifetime, but is really only examined as a footnote to Marx and for his impact on Continental philosophy now. He gave people what they wanted, and made it sound deep, but now it looks embarrassingly naive.
Sanatana Dharma thought is deeply woven with the idea time is cyclical, the breaths of Brahma & yugas, and that conditions now are getting worse, as we have entered the Kali Yuga. Pre-rennaissance European thought held similar views, looking to Classical golden ages and ruins they could not imagine building, and expecting the Day of Judgement any time. Whereas Hegel's dialectic was above all reifying progress, without limit.
