Person 1 used initially an argument that relied not on any specific properties of what we call "universe" but only on the words in a sentence. Basically he said
Everything in (X) (Y) therefore (X) must (Y)
for the special case X = "universe" and Y = "has a cause". No reason was given why this would specifically apply to this X and Y. Therefore his argument can be refuted by giving an example where it does not apply.
Person 2 proved the fallacy of the argument by demonstrating that the result of the argument is wrong in the case X = "wall" and Y = "is small".
By giving the counter example Person 2 demonstrated that what Person 1 tried to use as a logical rule was not a valid logical rule. If Person 2 had claimed (which he didn't) that because walls are made of things that are small but are not small themselves, the universe which contains things having a cause cannot have a cause, pointing out that the universe is not a wall would be a valid refutation of that claim. But nobody made that claim. We could therefore say that Person 1 was using the "straw man" fallacy: Refuting a claim that actually nobody made.
In reality, I'd say that Person 1 just didn't understand how Person 2's refutation worked. So his argument is probably just a non sequitur.
Maybe this needs more explanation: P1 used without any justification an incorrect logical rule, to deduce a result that would be hard to prove or disprove ("the universe has a cause" is obviously hard to prove or disprove). P2 demonstrated that the logical rule was incorrect by giving a use of the exact same rule that leads to an obvious incorrect result.
What P2 did was to demonstrate that P1 used a rule that wasn't generally correct. One could try to save some benefits of such a rule by finding conditions when it is correct to use. P1 didn't do this or even attempt to do this. One could try to demonstrate that a statement is correct without applying this rule. P1 didn't do this. Instead he just made the correct statement that "universe" is not "brick wall", probably trying to imply that therefore "universe" was not a counterexample disproving the validity of his logical rule, and probably trying to further imply that therefore his statement "the universe has a cause" was correct.
And that is a non sequitur. Of course this doesn't follow. All that P1 has shown that P2 hasn't given a strong argument that "the universe has a cause" is false. But it was P1 who made a strong statement and therefore needs strong arguments to support this statement, which he doesn't have.
Julian - the rule that P1 used is not a valid rule. Because it is not a valid rule, it cannot be validly used to argue anything. P2 demonstrated the invalidity of that rule.
P1's next statement just produces a response of "so what"? So if the universe is not a brick wall, what does that have to do with the fact that P2 demonstrated that P1 was using an invalid rule? Nothing. Nothing of any interest follows from his statement. "The universe is not a brick wall" as a response has the same quality as "a boiled egg is not a brick wall" - it proves nothing. And that's what's called a non sequitur - a statement that proves nothing (especially not what someone is trying to prove). That's the correct term for his fallacy - a non sequitur.