The company has announced a rights issue in the ratio of 1:15. If I have 10 shares of the company in my Kotak Neo account and 5 shares in my ICICI account, am I eligible for exercising the rights?

You will not get the credit of rights entitlements in this case. Since the RTA generates one folio against one demat account, and the allocation of rights entitlements happens folio wise, you will need to have the minimum number of shares, to be eligible for rights entitlement, in each of your demat accounts.