[Nero Germanicus]

College records don’t contain any information on where a person was born. My college transcripts list my address at that time and my grades, that’s it.

If anyone wants to know if Obama registered as a foreign student, the place to look is the federal government to see if he was ever issued a foreign student visa. (F-1 visa). You cannot attend school in the US as a foreign student without a visa and visa records aren’t sealed.
For example, Obama’s father’s student visa files were released under Freedom of Information Act requests.

[Leaning Right]

I'm thinking more of just transcripts here. It would seem to me that for financial aid reasons alone, questions about citizenship and place of birth would be asked.

Just for the heck of it, I checked the on-line application for a local college in my neck of the woods. They wanted citizenship info.