How can I replace the Forum's vocabulary 'Forums' with my existing vocabulary 'tags'?
After you enable the D7 core Forum module, you'll also have a "Forum topic" content type (and a "Forums" vocabulary as shown via admin/structure/taxonomy/forums).

Using the "Manage fields" tab of this content type (located at /admin/structure/types/manage/forum/fields), you can see the various fields for this content type. One of its fields is the "Forums" field (with machine name "taxonomy_forums"), with "Term Reference" as Field type.

Click the "Term Reference" link of the "Forums" field there, which brings you to /admin/structure/types/manage/forum/fields/taxonomy_forums/field-settings, where you can change its "Field settings".

There you will be able to change the (default) "Forums" vocabulary to (e.g) your existing "Tags" vocabulary (or any other vocabulary you have defined).

Attention: if on the "Field settings" screen the Vocabulary drop down is disabled (= you cannot select another vocabulary, as in the screenprint in the "PS" of the question), and you get an error message like "There is data for this field in the database. The field settings can no longer be changed.", then you'll have to think of a way to get around it somehow. A few more links that provide some more details on this (and possibly other alternatives):


Support issue ( = Error on changing field type: "There is data for this field in the database. The field settings can no longer be changed.").
Support request nr 1378894 ( = modifying "Field settings" with data in database).


Be careful when trying to get this fixed! Because things may get worse ...
You might consider a possible approach like this:


Create a new term reference Field for the Forum topic content type, and for which you then select the desired taxonomy.
After that, it should be OK to "delete" the original taxonomy_forums (machine name) field.


However, in doing so, the new term reference Field will have a machine name starting with a prefix field_. But if you then try to use that field (as part of the Forum module functionality), the Forum module might not use such  field_ prefix (you might get errors about that mismatch).

An alternative approach might be to uninstall/reinstall the Forum module, and then check if after enable you can change the taxonomy term field. However, in doing so, be aware that the taxonomy term "General discussion" of the taxonomy "Forums" (used by the module) is NOT deleted as part of the uninstall of the module. So before re-install (enable), you may want to try to manually delete that term and taxonomy first. If you don't do that, then you might end up with a taxonomy term like "General discussion" multiple times.