The institutions continued. The institutions continued under Augustus, too. But that's like saying that China and Venezuela are democracies because they have elections.
The Roman Republic was dead when Caesar won the civil war. The trappings remained, but the Republic was dead. (Seriously, could the people - or even the Senate - have voted Julius Caesar out? No, Caesar was there because of his armies, not because of the popular vote. So it wasn't a republic. In a republic, you can lose elections and be out of power.)
The Roman Republic was dead when Caesar won the civil war. The trappings remained, but the Republic was dead. (Seriously, could the people - or even the Senate - have voted Julius Caesar out? No, Caesar was there because of his armies, not because of the popular vote. So it wasn't a republic. In a republic, you can lose elections and be out of power.)