GPS _is_ less accurate vertically in certain circumstances (namely when you only have a lock on 3 satellites, as there are two possible altitudes you could be at), but usually this doesn't matter unless you're flying - otherwise just figure you're on the Earth's surface, and cross-reference map data for the altitude. Inside multi-storey buildings it's a different story, but you won't be getting a GPS signal indoors anyway.
GPS _is_ less accurate vertically in certain circumstances
GPS is less vertically accurate all the time, due to 'dilution of precision', which depends on the position of satellites in the sky [1].
When you move x distance horizontally, unless you have a very poor view of the sky some satellites will get closer and others further away. When you move the same distance vertically, all satellites will get closer but by differing amounts [2]. As vertical moves produce a smaller change in the measured data, noise in the measured data has greater impact on the vertical measurement.
just figure you're on the Earth's surface, and cross-reference
map data for the altitude
Once GPS data reaches an app on a phone, they're free to discard the height data, true.
But the GPS reception module itself doesn't contain a detailed elevation map of the earth; there isn't enough memory to store a detailed map, or any way to push out map updates.
usually this doesn't matter unless you're flying
The main application I've seen for height data is sports trackers. For example, cycling and running up steep hills is hard work compared to running the same distance on the flat. Some trackers will even set challenges like climbing 2000m in a month [3].
Things like drone altitude control also like having height data, obviously.
