How to identify which button was clicked

It's quite possible, but this is a bad idea. The code using such identification would be unstable, hard to maintain. This is not how event-oriented programming works.

One problem is: you did not tag the UI framework/library you are using. However, buttons are handled very similarly in all of them. Quite generally, the signature of the delegate added to the invocation list of the Click event instance takes two parameters: object sender and another one, of one of the event argument types.

It's possible to use the same method as the event handler for several buttons. Sometimes, it could make a good design. Apparently, this is what you do, otherwise you would not need to identify anything. Now, sender is always a reference to one of the button objects. So you can perform a set of checks: if sender = myFirstButton then… if sender = mySecondButton then… But this would be the abuse.

The general approach is to add a separate event handler to each button. In this approach, the handler should better be an anonymous function. I usually advice not adding any event handlers via the designer; presently, designers create obsolete code which is harder to maintain (if WPF, the situation is a bit different, but I still would recommend to be careful with XAML approaches), but it's really up to you.

As I say, using the same even handle can be useful, but only if several buttons do "almost the same thing". The rule of thumb is: whenever you need to "identify a button", you do it wrong. The right approach is: first, think why do you want to use the same handler? The answer will always be: because the code would be very similar. Now ask yourself, what would be the difference? When you figure this out, your problem is reduced to another problem: passing this "different item". You can use button tags, but the general solution would be, for example, the dictionary with button keys, and the values use in the handler logic. Hope you can grasp the idea.

—SA