While I have never spent any time in India, 'Today Morning' is common in Singapore, another Asian country once colonised by the British Empire and now claiming English as their Official/First language.
Also of interest:
I teach English as a foreign language in Indonesia. Here I teach many people whose first languages are either Mandarin, Indonesian or other local languages/dialects and a small number of European expats speaking various European languages.
This is quite a common error resulting from Mother Tongue Interference, whereby a student applies the rules from their own language to English and assumes a direct translation is correct.
Of course, direct translations are rarely valid due to the many unique nuances of a language.
"Today Morning" is a great example of this.
Another reason it happens is that learners over apply patterns they find in a language. This can happen with young native speakers just as much as second-language learners.
A learner sees that 'yesterday morning' and 'tomorrow morning' are both correct and so assumes that 'today morning' would be natural. Of course, we know that it is not. Another common example of this is forced '-ed' endings to past tense irregular verbs, such as 'drinked' where it should be 'drank'.
Hope that is of some use.