I think things are a bit more nuanced.
In the 90s, Azharuddin, Sidhu, Laxman, Tendulkar and Ganguly feasted on Sri Lankan spin. But Murali wasn't that good in those days, and those test matches were batting feasts for both sides.
People then point to Australia's 97 tour of India as being "proof of subcontinent players being good vs spin". Sure, Kumble bamboozled the Aussies and Warnie had no effect on India; India won that series comfortably. But I don't think India were really tested by spin in this series; Warnie had no help at all. All his big guns stayed at home.
And look what happened when Pakistan toured India soon after. Kumble murders Pakistan (didn't he pick up 10 wickets in an innings?), but Saqlain Mushtaq - more familiar with these conditions than Warne - absolutely knocked down all the Indian batsmen. Spin killed everyone, and the series only offered a lone, dogged ton by Tendy and some kind of epileptic fit of a ton by Afridi.
Other people say India's class vs spin really starts in the early 00s. They point to Austraila's 2001 tour where the Aussies lose even with McWarne. Yet Murali, now an ATG, totally bamboozles India some months later. Here Harbie Singh had no effect on the Sri Lankans, and Dravid and Ganguly couldn't score vs Murali.
Then, in 2005, a stronger India face a strong Sri Lanka, in India. Sri Lanka are clueless vs Harbie and Kumble, who spin them out, but the Indian batsmen are equally confused versus Murali, except Laxman and Tendulkar.
I think in the 90s and early 00s, aggressive sloggers had fluke successes vs spin in India (Hayden, Afridi, Sehwag), whilst the only consistently reliable Indians vs spin were Tendy and Laxman, who weren't exceptional, but dogged enough to outperform the nonsense foreign teams were doing vs Kumble/Harbie.