Custom theme being prompted for update because of similarly named theme

There is no intended way whatsoever in WordPress to opt out of updates. Even not intended way is significantly convoluted and needs to intercept HTTP requests being made.

You can disable updates completely with DISALLOW_FILE_MODS constant.

If you have to retain update functionality for other stuff you will have to code it from scratch or use a third party implementation, such as Update Blocker (disclosure — was created by me).