Was the democracy in ancient Greece the same type of democracy we have today?
No it was not the same. Democracy was first practiced in ancient Athens, not the whole of ancient Greece [which was not a country, but a collection of city-states]. Sparta for example, was an oligarchy not a democracy. Other smaller and less-powerful city-states did follow in Athens’ footsteps however. Only free Athenian male citizens over the age of 18 were allowed to vote. That meant only a small percentage of the population. Women, slaves, freedmen or citizens of other city-states were not allowed to vote. To be an Athenian citizen, both your mother and father had to be Athenians. The Athenian democracy was however the first of its kind in the world and the root of our own western democractic ideas.