non-capture group still showing in match

The entire match will always be group 0, you need to access that specific group (group 1 in this case since the first group is non-capture), you can do it like this:

var str = "<p model='cat'></p>";
var regex = /(?:model=')(.*)(?:')/g
var match = regex.exec(str);
alert(match[1]); // cat

Fiddle

Also, I suppose you are probably wanting several matches within str, you could do that like this:

var str = "<p model='cat'></p><p model='dog'></p><p model='horse'></p>";
var regex = /(?:model=')([^']*)/g
var matches = [];
var match;
while (match = regex.exec(str)) {
  matches.push(match[1]);
}
alert(matches); // cat,dog,horse

Fiddle

A non-capturing group is basically just a non-group ― a way to use parentheses without actually treating that part of the pattern as a group.

It looks like what you're actually looking for are the "match prefix but exclude" group (?<=) and the "match suffix but exclude" group (?=).

Note: This type of group does not seem to be supported in Internet Explorer.

If you use these, you get the desired result:

var str = "<p model='cat'></p><p model='dog'></p><p model='horse'></p>";
var regex = /(?<=model=')[^']*(?=')/g
var matches = str.match(regex);

console.log(matches);