First of all I’ve always emphasized that don’t start betting unless you have verified your account on any bookie site. The same is applicable to skrill as well. Once your account is fully verified then only proceed further to upload money.
This is a common practice, and security measure, for any bookie to verify your account dtails when you are withdrawing money. This is not a problem but a process.
e-Aadhaar card is not accepted as ID proof outside of India. You have to get passport there is not other way around it. If you want to live in cyber world then passport is must. You can apply passport under Tatkal scheme.
You can try driving licence, PAN card. Contact their customer service for further details.