Lets say for example I have a module "nospam". It's purpose is to check the content a user would submit in the site wide contact form (https://api.backdropcms.org/api/backdrop/core!modules!contact!contact.pa...).
For testing purposes, I have this function in my module:
function nospam_contact_site_form_validate($form, &$form_state) { form_set_error('message', t("Spam")); }
When I try to submit the form, I do not see the error above and the submission proceeds normally.
@BWPanda is correct.
If you want to validate a form created by other modules, you need to create a hook_form_alter() or hook_form_FORM_ID_alter() function and put something like what is in that validate link above in it.
Something like this untested snippet:
That might not be entirely correct, but it should get you on the path.