The answer is... its complicated... While there is definitely overlap between members of the Muj in the 1980s and the Taliban in the 1990s, the Taliban shouldn't be viewed as a successor group, as they were actually founded in opposition to the Mujahideen groups that took power following the ouster of the Communist government in 1992. This previous answer I wrote goes into it more in-depth, but I'd be happy to answer any questions you have from it here!

You should check out this answer here.

Also "Taliban" has a very generic use in Afghanistan. In Dari, talib/taliban just means volunteer/volunteers. Many people who during the cold war would have described themselves as taliban did not later join the group to whom the west refers as "The Taliban".

The US and Saudi Arabia did arm the mujahideen during the 80s, yes, and some became was is known as the Taliban. There were a lot of different resistance groups in Afghanistan, and then many more Arabs came from Pakistan, Yemen, Oman, Palestine, Lebanon - you name it - to expand the war into a jihad.

The Taliban was formed by students who thought that by expelling the invaders and purifying the religion, relying somewhat on takfirism, or declaring those who didn't believe in their brand of Islam as heretics, they could establish an Islamic caliphate. The US didn't really care about their religious crusade like Saudi Arabia did, but they did pump money and weapons into the country.

The Looming Tower by Lawrence Wright is a really good introduction to this. Won the author a pulitzer prize.