I made a CPT for "Events" and another for "Notices" but the Events' categories are showing up in the Notices' categories. Shouldn't those categories be totally different? How can I get them to have different sets of categories?
Please understand that the Plural/Singular name of the taxonomy can be the same but the slug is different. WooCommerce product category has the same Plural name as the Post category but has a different slug product_cat. You can see it in the URL https://monosnap.com/file/oO8j6PCtT9wMyTDlvwyGDxoDXgT4EH
You can change the Notice category slug to notice_cat by going to Meta Box > Custom Taxonomies > Edit Notice and change the slug. The taxonomies can have the same Plural/Singular name but must have different slugs, the slug must be unique. You can read more here https://developer.wordpress.org/reference/functions/register_taxonomy/
and the section Reserved terms.