Your application is bound to a specific user by virtue of the OAuth Access Token it submits. So provided your app can retrieve an Access Token for User-2 (with an appropriate scope), then you're good to go.
Getting an access token for User-1
There are number of ways you can do this, depending on the volatility and security requirements of your app. Fundamentally it requires your web app to have a stored Refresh Token for User-1, which it can use at any time to silently fetch an Access Token. See How do I authorise an app (web or installed) without user intervention? (canonical ?) for more details.
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Some of the terminology in your question needs a bit of straightening out.
- "ID" - each application has its own ID when you register it on the developer console. It's refered to as a Client-ID which confused some people into thinking the ID is somehow related to the end user; it isn't.
- "Session" - there is no session to Google Drive. There may be a session to Google, within which the OAuth flow will seek to identify the Google user in order to generate an access token for that user. But there is no "session" to Drive, which is why a Drive app can access as many Drive accounts as it likes, provided it has an access token for each one.