It sounds like you’re creating both the consumer key and secret (i.e. the app identifier) and the account access key and token (the user identifier) in the apps.twitter.com dashboard.
The latter is not the best idea in this case, as it means that anywhere the combination of those four values is known, the app will have access to YOUR account. It is useful for testing, but generally speaking, you should not share the account access key and token.
What you want to do is to create an app, get the consumer key and secret, and then implement Sign-in with Twitter aka three-legged OAuth. Once you’ve done that, your app will receive the access token and secret for the newly-logged in user, and you can store and re-use that when you make calls on behalf of the third party user. That’s how multi-user authentication through a single app works on Twitter.