There is a short, simple, and clear answer to this: No. Different Sects within Hinduism do not fight and have not fought each other based on religious differences. In fact, belief-based differences between different sects of Hinduism are very small and rare. Most differences between the sects of Hinduism are practice-based.
Hinduism is a polytheistic, decentralized religion. We have no central authority and there is no "king" of Hinduism, like how Christianity has a pope and how Islam had a Caliph. We are also encouraged to have multiple gods and deities to pray to. (There's more nuance to it: For more details see the answer to [this] post). Vaishnavites spend more time and dedication towards Vishnu, but the accept and revere Shiva. Shaivites spend more time and dedication towards Shiva, but they accept and revere Vishnu. Their beliefs are the same, but their priorities and practices vary a little.
A common notion amongst Hindus is that the same god can have many names. For example, [Ganesha] has other names such as Ganapati and Vinayaka, but also many different tribal names for him.
So, there were and are no hate or war between Shaivites and Vaishnavites.
Why didn't Hinduism schism? The reason is simple. Centralized Religions typically Schism, while decentralized religions typically never schism. For example, Islam had a schism between Sunnis and Shias, because of a dispute of the "throne", or position of Caliph, of Islam. There is a possibility to gain power through the ladder of the centralized religions, and that leads to disputes, which lead to Schisms.
Christianity had a Schism between Catholicism and Orthodox Christianity because Christianity was a highly political force during the time. Politics leads to greed for power, and that leads to disputes, which leads to a Schism.
Meanwhile, Hinduism is a highly decentralized religion. We have no leader. This leads to no possibility of using Hinduism as a way to gain power. So, there is no power conflict, which means there is no schism.
Another reason why there was no schism in Hinduism is because, in the outdated caste system, kings have less importance than saints, which means that when a king tries to defame the other sects of Hinduism, no one will listen to him since they know that the king is not the highest authority on Hinduism, the saints are. And since saints are decentralized, there was never a schism.
Hope this makes sense! Feel free to edit this to help me.
: How did Ram and Krishna become the dominant Gods in Hinduism?