McGrath said: "I'm still puzzled by why so many people seem to think that beliefs about religion have some intrinsic relationship to opinions about politics."

Because there are often similar thought processes and basic logical syllogisms involved. From a purely technical, philosophical point of view, an analysis of the form of many of such beliefs & opinions reveals similar structure. Of course it is not always the case.... but in some instances there are ways in which data is presented, authority cited and defenses mounted that use rhetoric to evade direct confrontation with strict rules of presentation. (Yes... we are all guilty occasionally).
This does not necessarily imply dishonestly or evil intentions.... it can just mean that someone learned a cultural or family bias, and automatically start with the assumption that it is 'truth'. (I read a quote once that said: (paraphrased) "Theology is the formal process of finding bad reasons for what you already believe."
   In politics the point was made in a joke about a Democratic politician giving a stump speech and being interrupted every few sentences by a guy yelling "I'm a Republican!". Finally the speech maker stopped and looked right at the offender... "And just WHY are you a Republican?..."Because, my daddy was a Republican, my granddaddy was a Republican, my great granddaddy was a Republican... and I'll always be a Republican!"
"So," asked the politician, "If your daddy, your granddaddy and your great-grandaddy had been jackasses, what would YOU be?"
   "Oh," said the heckler, "I'd be a Democrat!"

The point is, many people don't need or want clear, unambiguous logical reasons for what they believe... some things just 'feel' right, and they often can't begin to tell you exactly why they think that way. IF we could get their total family history, personal history, school history and a detailed psychological profile done by a team of psychologists, we 'might' have some idea of how & why they think as they do... and it would do little good in most cases.
In the awkward human process of learning to think, it takes a real effort to, as Socrates and others said, "Know thyself". Examining every decision critically is never easy.... and even running important decisions about religion & politics & science thru fact checking and logical analysis is not common. Those who DO make the attempt more often are just... different... from those who do not.... and that is why is is said that "beliefs about religion have some intrinsic relationship to opinions about politics".

It is a lot of damned work to even explain the issue in an succinct way..(as those who have read this far will agree)... and WAY too much work for "Joe Average" to follow Socrates' admonition. He'd just rather be a jackass... I mean, Republican...