Might depend on what exactly one considers a point of interest, but looking at some common keys:

• http://taginfo.openstreetmap.org/keys/amenity
• http://taginfo.openstreetmap.org/keys/shop
• http://taginfo.openstreetmap.org/keys/historic
• http://taginfo.openstreetmap.org/keys/tourism
• http://taginfo.openstreetmap.org/keys/craft
• http://taginfo.openstreetmap.org/keys/office
• http://taginfo.openstreetmap.org/keys/leisure
• http://taginfo.openstreetmap.org/tags/natural=peak
• http://taginfo.openstreetmap.org/tags/highway=bus_stop
• http://taginfo.openstreetmap.org/tags/railway=station
• http://taginfo.openstreetmap.org/tags/aeroway=aerodrome

we get around 20 million features that could be potential landmarks or points of interest.

Just counting points that have names, we get around 12.9 million features.

Addresses

• https://taginfo.openstreetmap.org/keys/addr:housenumber
• https://taginfo.openstreetmap.org/keys/addr:postcode

70 million individual addresses and 49 million postcode locations

Maybe there is a better way of finding out, but here’s a new conversation starter at your next mapping party!