For most Americans, the stars and stripes is not a sign of temporal power, i.e., an expression of the authority of the federal government. This is something I've thought quite a lot about since joining FOTW but I'm still not sure I've sorted it out, so bear with me. As I see it, the display of a national flag can be intended to convey several distinct messages. The same display may express more than one of these messages. However, different countries tend to emphasize these messages in different ways, sometimes emphasizing one interpretation to the exclusion of others:
The person flying this flag is acting under the authority of the sovereign the flag represents. (People who are not representatives of that authority (i.e., private citizens) have no right to fly this flag; "don't you dare fly the Union Jack at sea unless you're one of HM ships!")
The place where this flag is flying is under the authority/sovereignty of X. Flags representing other sovereigns may not be flown. ("Just because Slobovian air forces are flying out of a base in Ruritania doesn't mean they can fly the Slobovian flag; that would be an assertion that it's their base, not Ruritania's.")
The person flying this flag owes allegiance or obedience to the sovereign the flag represents. Its display is an expression of loyalty or submission.
The person flying the flag wants to show respect, hospitality, genealogical connection, linguistic facility, etc., etc., having to do with the country the flag represents. ("The Polish flag in the chancel of St. Stanislaus Church represents the ethnic origin of its founding members.")
In the United States, people tend to gravitate to messages 3 and 4. I don't mean that this is anything one does consciously, it's just part of the culture. Flag laws and practices elsewhere are more consistent with messages 1 and 2. In other words, an American Baptist congregation that puts the S&S inside the church is not asserting the government's temporal power over the church (message 1) but expressing the allegiance that the members of the church as U.S. citizens owe to the country. Likewise, a Catholic church that displays the papal flag or a synagogue that displays the flag of Israel is not asserting Vatican or Israeli territorial sovereignty over the property (message 2), merely religious allegiance (message 3) in the first case and ethnic or political sympathy (message 4) in the second. In a different cultural context, all these messages could obviously be misinterpreted.