Grass is native to lots of places, lawn that's meticulously kept clean of every other plant species? Not so much.

The meme doesn't say that grass isn't native, but that they want to replace it with native plants. I do get your point though, it sounds like it excludes grass. According to Wiki, Lolium perenne (hugely cultivated grass worldwide) "is native to Europe, Asia and northern Africa". I think why people explicitly say that they want to replace their lawn with native plants is because less knowledgeable people might go with introduced and widely cultivated species. This would not be as effective in bringing back wildlife. Introduced plants are often not great host plants for pollinators for example. So it is great when people specifically replace their lawn with plants that are native to their region.