Is it possible to make OpenStreetMap work like Wikimapia?

Help and support
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The closest thing on the main site today is the Map Data layer, which you can enable from the Map Layers sidebar:

A screenshot of OpenStreetMap.org with the Map Layers sidebar open on the right. An arrow points to the button that you click to open the sidebar, and another arrow points to the Map Data checkbox in the sidebar. The checkbox is checked, so the main map has an overlay of all the raw data in the database, including streets, buildings, and sidewalks.
A screenshot of OpenStreetMap.org with the Map Layers sidebar open on the right. An arrow points to the button that you click to open the sidebar, and another arrow points to the Map Data checkbox in the sidebar. The checkbox is checked, so the main map has an overlay of all the raw data in the database, including streets, buildings, and sidewalks.1920×1073 248 KB

With this overlay enabled, click any of the elements highlighted in blue to see the details about that element. This isn’t quite comparable to what Wikimapia exposes, since the OSM data model differs from Wikimapia’s, as @Kovoschiz explains.

Yes, this would be trivial for any map powered by vector tiles and a library such as MapLibre GL JS or Mapbox GL JS, using the maplibre-gl-inspect plugin.

For example, in response to a discussion about abandoned railways, I whipped up a quick demonstration of combining OSM and OHM in the same map. OHM isn’t relevant to this discussion, but I happened to install maplibre-gl-inspect with one line for debugging purposes. Thanks to this plugin, you can click the button in the upper-right corner and hover over each feature to see its properties according to the vector tiles:

A map of The Banks neighborhood of Cincinnati, rendered as translucent polygons color-coded by thematic layer, with a callout box enumerating the properties of Central Riverfront Garage.
A map of The Banks neighborhood of Cincinnati, rendered as translucent polygons color-coded by thematic layer, with a callout box enumerating the properties of Central Riverfront Garage.1920×1073 104 KB

Unlike the Map Data overlay above, the vector tiles apply some transformations and filters to OSM data, so take the specific properties with a grain of salt.