The Lighthouse building in the Harbour says it gives +2 housing for cities adjacent to Coast. Does this include cities built beside a Lake tile (but not beside an ocean Coast tile)?
Coast is a tile type in Civ 6 that you can check by hovering over the tile. Coast tiles are usually found in between Ocean and land tiles, not Lake and land tiles. Lake tiles are typically adjacent to land or other Lake tiles.
However, depending on the map type, small oceans can be generated that behave and look like multi-tile lakes. This has happened to me on Fractal maps.
So, you need a real Coast tile adjacent to the city to get the +2 housing. Check the tile type by hovering over it.