Installing a Bottle Cage without Mounting Screws/Threads

The approach I've seen is to use hose clamps to hold a bottle cage to the frame. Hose clamps are what (used to?) attach a car's radiator hose to the radiator, if you've seen that. If you have an old inner tube, you can slice that into strips to pad between the clamps and the frame so the paint doesn't get scratched.

The answer of @compton brought me an idea - hose clamp but nicer - hose wire binding, like this: With some pieces of inner tube it will not damage the paint job and with some isolation tape it will be almost invisible. The reasonable place for the knot is the underside of the frame so not disturbing anyhow.

Here is how to use one (DIY):

And here is how to make one:

Here are a few that I found while looking. easy if you know what to look for.
Will depend on what you want and need

I've used rivnuts to create mounts for bottle holders. It requires you drilling into your frame and then tightening up the rivet nut, but once you have done that you would have effectively the same mounts that are present on bikes with the cage mounts pre-installed.

You would probably want M5 size rivnuts for compatibility with other bottle cage/fender bolt thread sizing.

Beware This may compromise the strength of your frame. Don't do this on carbon fibre or thin walled metal frames.