So the Golden Horde was not Islamic? Not even the Crimean Khanate?
.....Are you aware of the demographic history of Crimea after the Russian conquest, and the reason why most people there speak Russian today while there are a million Crimean Tatars in Turkey? Ditto, are you aware that Tatarstan is currently majority Muslim, and that the large Orthodox minority is largely composed of Slavic settlers? The Volga Tatars are almost entirely (as in 99%) Muslim, with the sole exception of the
Kryashen Tatars. And a huge number of the Kryashens were crypto-Muslims all the way until 1905. That year the tsar made concessions to Islam as part of the aftermath of the 1905 Revolution, and an enormous number of the Kryashens immediately went back to publicly practicing Islam.
Sindhi, Punjab(all of it), Kashmir, Gange. All totally animist areas?
Sindh is an atypical case where the Arabs conquered it early on. You are definitely wrong about all of Punjab being Muslim (like why do you think there's an Indian Punjab in the first place?) and those parts that converted were discussed above. The history of Islam in Kashmir is debated, but it seems that it was an indigenous development like in Southeast Asia, so of little relevance to OP's scenario. It also happened rather before my time frame. I have no idea what "Gange" is supposed to mean, but the Gangetic Plain was almost entirely Hindu.
because if you are able to convert 2 digits percent of the population, then most of India would be included
You have a generous definition of "most," especially in the aftermath of six centuries of Islamic rule. Also, the Muslim proportion of the population actually rose a lot under British rule. In 1826 the British estimated that only 1 out of 6 Indians were Muslim, while in the 1940s it was 1 out of 4. The vast majority of the Mughal empire (whose Muslim population is usually estimated at around 10%, accounting for the smaller populations of Bangladesh and Punjab) would have a vast Hindu majority.
Also it wasn't portion of the Bengal that was converted, it was pretty much most of it, even if it was not majority everywhere.
No, only Bangladesh, most of which was jungle until the Ganges changed course just as the Mughals were strolling in. The old pre-Mughal capitals are either in now majority-Hindu areas (Murshidabad, Bankura, Gauda, Nabadwip, Pandua) or in places that were only 50~60% Muslim before partition (Sonargaon, Naogaon, Munshiganj). The most Muslim areas are in the northern and eastern parts, which are not major centers of Bengali civilization and some of which weren't even part of India until the Mughals conquered it (Chittagong).
I'd like your opinion on why this is, if Hindus and animists were equally likely to convert.
Dominated by animist? Those were Christian, syncretic or not.
Islam spread under the auspices of the rulers of Sinnar, who were basically animists until the 18th century and never followed Christianity. Christianity was in a general state of collapse in Nubia
before the advent of Islam, and animist pastoralists -- who, remember, had greater prestige than agriculturalists, as is generally the case in Africa -- had made huge inroads already.
Islamic control of India did lead to conversion to Islam, Islamic control of the Balkans also did.
If it's identity, what does this rule that "Religions conversion of world religion post 1500 = impossible" come from?
Because Abrahamic religions and (at least) Theravada Buddhism are transcendentalist religions who tend to become a central marker of identity among a population, to a far larger degree than other religions. This process was exacerbated in the Early Modern era, by which all major world religions had more or less assumed their current orthodox form (hence why your example of the Arab conquests doesn't really work -- early Islam appears to have been a very amorphous thing) and, especially in the case of Christianity and Islam, had an explicitly oppositional attitude towards Islam and Christanity. If you have JSTOR access, read Alan Strathern's
Transcendentalist Intransigence: Why Rulers Rejected Monotheism in Early Modern Southeast Asia and Beyond.
Islamic control of India did lead to conversion to Islam, Islamic control of the Balkans also did.
Why was there not a majority in either areas after six centuries of Islamic rule for the former, and four to five centuries in the latter?
Let me reverse the question, in which centuries long control of one world religion over another didn't lead to conversion to the former of sizeable amounts of the population?
Well, virtually every sizable Muslim, Hindu, and Theravada area ever conquered by Europeans (except where the old inhabitants were actively chased out or swamped by immigrants, like the Russian steppe or Andalusia)? Not to mention most of India, despite undergoing Muslim rule for more than six hundred years. The Ottoman Balkans.
Actually, look at a map of world religions in 1500 and the same map in 2000. You will see that Muslim-majority territories in 1500 are almost all Muslim-majority in 2000, with the only big exceptions being Spain (where the Mudejars were forcibly converted), Palestine (where the Arabs were kicked out), and parts of the steppe (where the actual Tatars are still almost all Muslim). Similarly, you will see that all Christian-majority territories are Christian-majority in 2000 with exceptions in parts of Turkey and the Balkans, and maybe parts of northeastern Africa (also ethnic change, with Nubians a small minority and the Oromos expanding). If you look at a map comparing the extent of Theravada Buddhism in 1500 and 2000, they will nearly match, the main exception being in southern Vietnam where the Vietnamese aggressively conquered and assimilated the indigenous Cambodians.