If 'an' is required to be used with a vowel sound, then why don't we say an Universe, an Unicorn, although I know it sounds Y in the beginning. Then Why U is part of the vowel in the first place.

We pronounce u as [ju:]and it sounds Y in the beginning.And Y is a consonant
sound that is why required to be used with 'a'.We say a universe, a union, a unicom.

Thanks for replying, but as I see you are just repeating me. My question is: when pronouncing 'University' sounds like 'Yu' and so 'a' and not 'an' should precede it, then why should 'U' be part of the vowels in the first place.

There are times when u acts as expected (i.e., exactly like a vowel) and times when it acts like a consonants. For example:

a unicorn
an unintended benefit

It happens with consonants too:

a house
an hour

The letter u most often acts like vowel. It only acts like a consonant at the start of some words. That's not enough to drag it into the consonant category.

Here is more on using an and a.

Thanks Craig.