Can I set a JS variable to a Twig output tag?
In your first example, you are outputting an ElementCriteriaModel. That is a query for tags rather than the tags themselves. It doesn't output anything usable by JS You're almost doing the same in the second example, except that your reference to gameTags in the JS is just referencing itself rather than any Twig output.

What you want to do is to query the tags and output in the way that JS can use. Something like this:

&lt;script type="text/javascript"&gt;
    var gameTags = [{% for tag in craft.tags.group('gameTags') %}"{{ tag.title }}",{%endfor %}];
&lt;/script&gt;


That should output a JS array of strings with each string being a tag. Something like below.

&lt;script type="text/javascript"&gt;
    var gameTags = ["action","adventure","platform","rpg","sports",];
&lt;/script&gt;


If you're a stickler for commas in arrays, there's ways to not output that after the last tag.