Does each country have their own Euro coin?

Each country makes coins and Bills and releases them in their countries. But they’re valid in the whole eurozone.

The front side of the coin looks the same, only the back is different for each country. There are also commemorative 2€ coins celebrating events or other things.

The bills look all the same you can only tell the issuing country by the first letter of the serial number.

2 euro commemorative coins - Wikipedia

Sounds like how in the USA there are different quarters for each state and special editions for anniversaries and stuff but bills all look the same
Yes, even Monaco or the Vatican get their own euro coin.

They each have many, as they regularly release new ones. And they soon percolate to the rest of Europe.

It’s always fun trying to figure out where a coin came from.