X's argument seems to contain the implicit assumption that if something is not explicitly mentioned (or implied) in the Bible then it must not (or more charitable, probably does not) exist. This assumption is clearly suspect. Essentially, X's argument is the one committing the fallacy of appealing to ignorance:
The assumption of a conclusion or fact based primarily on lack of evidence to the contrary. Usually best described by, “absence of evidence is not evidence of absence.” - logicallyfallacious.com
Just because there is no evidence of aliens in the Bible, doesn't mean they don't exist. There are several ways one could further respond. From a theological standpoint, I'd argue that it's implausible to assume that God told human beings everything there is to know in the Bible. The Bible is a book meant to communicate the story of God's redemptive process through history for humans, not a cosmic encyclopedia of all knowledge.
Now, you say that it seems Y is making an appeal to ignorance. I disagree. Notice Y is simply saying X is unjustified in believing the proposition that "because the Bible doesn't mention aliens, they don't exist" because his assumption which I stated above is false. Y then offers counterexamples. There is nothing fallacious here. Y isn't saying, because X has no evidence for his belief, then the converse belief is automatically true.
In another vein, one could argue that the entire discussion between X and Y is fallacious because they're both taking for granted the proposition "The Judeo-Christian God exists", as I've seen some point out. Again, I would have to disagree. The reason it is taken for granted that God exists is that the discussion between X and Y implies both X and Y are Christians. Thus, they both already accept that God exists. If both parties in an argument agree to take certain propositions for granted (as X and Y seem to implicitly do), no fallacy is committed by either party by assuming the truth of the said proposition.
The fallacy only comes when one party in an argument takes for granted a proposition that has not been argued for or the opposing party has not agreed to accept. If X was a Christian and Y was an atheist, then the discussion between X and Y would be fallacious since X is assuming that God exists in his argument, which is a point Y would contest.