The idea that you could be required to identify yourself is probably what seems strange to them. In (at least) several US states the police can require you to identify yourself, but telling them your name is all that is required. Americans have an aversion to "papers please", because for decades that is one of the things that we felt distinguished ourselves from certain other countries.
Quite that is why the UK drooped ID cards after WW2 rather surprised that other countries which had much more bad experience with "papers please" in WW2 did not do the same.
Visiting the US for a while, I recall being yelled at by an inland border patrol guard in Texas for having the temerity to keep my passport in my trunk. Saying my name was definitely not all that was required.
Land near the borders (particularly the southern border) is treated differently (very poorly, as you unfortunately learned). Texas is not even the worse in that regard, as hard as that may be to believe. Arizona takes that shameful crown: http://en.wikipedia.org/wiki/Arizona_SB_1070
