What is happening is that the source of the filters is the collection, not the tags. The Filter Menu app will try to keep the collection and the tags in sync, so if you manually add a tag it will see that the newly tagged product is not in the collection and will remove it. This is a good thing though, as it is often much easier to manage your collections with rules rather than having to manually apply many tags to each of the products.
So at the end of the day what you need to do if you want to add a product to a filter is to either:
1) Adjust the product properties such that it appears in your desired collection2) Adjust the collection rules, or if it is a custom collection add the product to the collection.
Also check to make sure your new tag does not match the rule set in the Variant Tagger, if you have that enabled. This will have the same effect of removing any tags you add, as it tries to keep your variants and the tags in sync.