Drill down option is only supported in report page, but not in dashboard currently. Here in a similar idea shared on Power BI Ideas. You can vote it up and add your comments there to improve Power BI on this feature and make it come sooner.
Q.1 - I have shared the desktop with some other person. But he is unable to perform drill down on charts and I don't want to share it as web url because that is a public url.
Is there any solution to this problem or Is there any option to enable it??
If Publish to Web isn't an option here, I don't aware any better solution currently.
Q.2 - My second question is that If I want other person to see the dashboard on mobile app, so Is it necessary that other person should have a power bi license? I mean from which login id he will login?
They need to use the same account that they signed up for Power BI service to view dashboards in the Power BI mobile apps. For more details, you can refer to this article.