"That is he" is technically correct, and is probably the answer you want to give on tests asking this question.
"That's him" is far, far more common in speech-- I believe this is true across the board in English variants. In AE you also see it a lot in writing. It would seem to me that in my lifetime the distinction between "linking" verbs and transitive verbs has broken down-- they both take objects and/or objective-case pronouns.
I would add that in AE "that is he" sounds goofy, pretentious-- it will get a "who's de perfessor" reaction.
What is the difference between a linking verb and a transitive verb? I would say "I gave the book to him", not "I gave the book to he". The object of the action should be in the form of "him". So if I say "That is he", I must conclude that this is not objective.
A linking verb equates or attributes one thing to another. In the first place a noun (or pronoun) to a noun, and in the second to an adjective.
1. He is the rascal I was talking about.
2. My efforts to reform are pointless.
When the subject of a sentence acts on an object, then a transitive verb is used. In a sense it subordinates where the linking verb equates or draws similarities.
1. He violates too many rules of conduct. (that's why he is a rascal).
2. My self-improvement plan yields negligible results. (that's why my efforts are pointless).
So noun-attribute linkage and subject-object linkage are similar, and sentences of either sort can be constructed to mean similar things.
The point is, when it comes to pronouns, people make statements linking two things, and they use the objective case universally, ignoring the (grammatically correct) distinction between (transitive) verbs that take the objective case and (linking) verbs that link to a "subjective complement" rather than an object, and therefore take the nominative case.
You are correct, titan2.
As crazy as it seems in this world of English, the linking verb is should take the subjective complement and should be in the nominative case....
....It's just not done in regular conversation--and not done a lot in written dialogue.
As foxfirebrand points out, if it's for a test, use the nominative case pronoun after the linking verb.