Not quite. Although the Republic is in the EU, it's not in the Schengen open-borders zone. (The UK isn't either). This means that anybody flying to Dublin from elsewhere in the EU has to pass through passport control. Existing EU citizens have automatic right of entry: but anybody else only gets in if the Republic wants to let them in.
There's a separate bilateral Common Travel Area agreement with the UK: so that arrivals from the UK, alone among EU countries, don't have to pass through passport control.
...or more accurately, didn't rather than don't. Despite the Common Travel Area, there's been an increasing tendency in recent years for the Irish authorities to require ID from arrivals from the UK.