Poi Comparison across Europe

You can probably start by having a look at the Wiki, especially at the map features to see how we map the most common feature. Then you can look at TagInfo to find out how often certain tags are used, but you will also be able to find undocumented values for e.g. shops.

But as SomeoneElse wrote, quantity is a lousy way to compare data sets.

OSM uses free tagging, so there is no "complete" list of tags available. You can see and search tags that have been used previously at https://taginfo.openstreetmap.org/ .

The answer to the next question "how complete OSM data is compared to other providers" will vary both by the thing that you're interested in and the location. Some area of Euope are extremely well mapped - pretty much every shop opening and closing will be noted.

The other side of the coin is how up to date the commercial providers are. I only have experience of HERE, Garmin and Google, and they each suffer from out of date data or the "fake locksmith" problem (people being able to game the algorithms used by Google et al to add data). Fake POIs do exist in OSM, but tend to stay around for less time, perhaps because the barrier to removal is less.

If you can give a bit more detail - what sorts of POIs are you interested in and where? Then people might be able to give more specific answers.