You will need 2 plugins, one to show the HTML and one to process what to do with the user input:
1) Make a plugin that fetches the pages, maybe the hook user.component_controller_register_1
or another in that controller would do but otherwise you can always use a low level hook like run_start
and check if its the section where you want it.
To show the HTML you have 2 options: include a JS file to populate the sign up form with anything that you want, or if you have a custom template you can just assign the array to the template variable and look through it in your template.
2) Once the html part is showing and working make a plugin for the sign up routine, I think the hook user.service_process_add_1 should be enough given its location.
Dont forget that the input name for the signup is an array, so your drop down needs to look somewhat like this:
<select name="val[my_dropdown]">