Yes, there are unicorns in the Bible.
||KJV Occurrences of "unicorn" or "unicorns"
||God brought them out of Egypt; he hath as it were the strength of an unicorn.
||God brought him forth out of Egypt; he hath as it were the strength of an unicorn: he shall eat up the nations his enemies, and shall break their bones, and pierce them through with his arrows.
||His glory is like the firstling of his bullock, and his horns are like the horns of unicorns: with them he shall push the people together to the ends of the earth: and they are the ten thousands of Ephraim, and they are the thousands of Manasseh.
||Will the unicorn be willing to serve thee, or abide by thy crib?
||Canst thou bind the unicorn with his band in the furrow? or will he harrow the valleys after thee?
||Save me from the lion's mouth: for thou hast heard me from the horns of the unicorns.
||He maketh them also to skip like a calf; Lebanon and Sirion like a young unicorn.
||But my horn shalt thou exalt like the horn of an unicorn: I shall be anointed with fresh oil.
||And the unicorns shall come down with them, and the bullocks with the bulls; and their land shall be soaked with blood, and their dust made fat with fatness.
However, the "unicorn" spoken of in the Bible is not the mythical creature of modern folklore: it is most probably the rhinoceros, but some doubt exists as to the exact meaning of the Hebrew word.
As for the example question in the tour, it obviously has nothing to do with the Biblical concept, or even with the Bible or hermeneutics at all. And the answer provided is just as frivolous as the question.
It could be modified slightly, both question and answer, while still retaining the unicorn theme. Or it could be replaced entirely with something else.
If one wishes to maintain the "unicorn" theme, I would propose the following:
Q: Did the unicorns mentioned in Psalm 22:21 really exist?
A: Yes; however, the Biblical use of "unicorn" does not match the
imagery of the modern fairy tale. Some think it refers to an extinct
type of ox, but it is also quite possibly a reference to the
rhinoceros. The Bible characterizes the unicorn as having strength
(Numbers 23:22; 24:8), horns (Deuteronomy 33:17; Psalm 22:21), and a
will of its own (Job 39:9-10).