Why not make $error_message an array?

This is an entirely valid way (i.e. assigning all errors to an errors array). In fact, I have an error manager class for some projects at work (e.g. reading/manipulating files, ensuring they are in the correct format, correct headers, etc) where I do exactly this.

With this class I also have various levels of errors such as fatal, warning, etc. Then, when applicable, the process checks for existing errors and when errors are found they either abort the process (fatal) or give the user a warning while still being able to perform an action.

You will have to store the array someplace where it can be persistent such a global or class level variable and push each error to the array to be retrieved at some point later.