Does a city by a lake count as coastal for the Lighthouse?

Unlikely!

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.