I downloaded a login form but I'd like to change its content without actually changing the plugin files so that it won't cause problems when updating later on. How can this be done?
There are no filters for the plugin's output and this plugin wasn't built as a class (where you could technically extend the class and override the parent methods).
If you're wanting to change the front-end elements only, you can use jQuery to do your bidding. If you need to add additional processing on the backend, you're out-of-luck with this plugin unless you fork it.
If you're intent on changing the plugin, I would create a GIT repo to track the changes of the original with your edits. GIT will identify the diffs for you.