If you are asking about vector maps, I am not really sure, because I have no experience with it. So can’t really comment on that. On raster maps, as you already know every tile is a PNG. The behaviour you described is very similar to the client side caching that usually happens in the browser. Depending on the coordinates in the viewport and zoom level the server provides the tiles.

Usually to save the map most offline map making tools will ask you to draw a rectangle and select the required zoom levels. In an interactive map, the rectangle is the viewport of the device. So there can be a feature which will download and store the tiles around a specific gps location for a fixed geographical area. That should be doable without much issue. But in this case that may not be a good idea.

If you visualise all zoom levels stacked over each other, the images need to be retrieved when the user zooms into a point the geographical area will not stay the same. Smaller geographical area is only needed with higher zoom levels. If we only take all the tiles that get downloaded in every layer, it may produce a shape similar to an inverted pyramid. So saving the images as a user zooms in for the first time, may be the best idea.

Then the saved tiles need to be used again when users zoom in the same area. Also these tiles need not be updated frequently and maybe even once in every 3 months might be enough, that too only when the user zooms in again in that area.

This can be a little tricky as almost all the tools that create offline maps do it for a fixed area and selected zoom levels, every point in that area gets equal priority. But in this case the point is the important element. The area nearby may not be relevant at all. So that is the part that needs some exploration.