Mormon Prophet Joseph Smith claimed that God the Father and Jesus Christ appeared to him. How did he know it was really the Father and the Son?

In regards to the statement according to the Bible God the Father cannot be seen.

John 1:18 has a JST which provides clarification (see @Gisheri's answer for examples of people seeing God)

19 And no man hath seen God at any time, except he hath borne record of the Son; for except it is through him no man can be saved.

John 6:46 also reflects a similar idea to the above

46 Not that any man hath seen the Father, save he which is of God, he hath seen the Father.

1 Timothy 6:16 also has a JST

16 Whom no man hath seen, nor can see, unto whom no man can approach, only he who hath the light and the hope of immortality dwelling in him.

In regards to the statement 2 Corinthians 11:14 it makes reference to angels of light. Joseph Smith did not see an angel of light, but God the Father and His Son Jesus Christ (the masters not the servants).

Joseph Smith could've/might've asked to see Jesus' marks like Thomas. Jesus could've offered. Jesus or God could've revealed further evidence so that Joseph Smith would know who it was that appeared to him as his account mentions:

20 He again forbade me to join with any of them; and many other things did he say unto me, which I cannot write at this time.

However using John 20 seems an interesting choice, as I see the verses as more of a rebuke to Thomas for asking when he knew/should've known Jesus was the son of God, not a pattern to be followed.

_{all emphasis mine}

I apologize for the long quotation, but Joseph Smith's explanation of the event is helpful for someone to gain an idea of why he (or anyone) may have been thoroughly convinced. As a 14 year old boy having an experience of this magnitude, pausing to question the personages on their authenticity was probably the last thing on his mind. Especially since they'd just delivered him from what felt like destruction after he'd called upon God with all of his might.

15 After I had retired to the place where I had previously designed to go, having looked around me, and finding myself alone, I kneeled down and began to offer up the desires of my heart to God. I had scarcely done so, when immediately I was seized upon by some power which entirely overcame me, and had such an astonishing influence over me as to bind my tongue so that I could not speak. Thick darkness gathered around me, and it seemed to me for a time as if I were doomed to sudden destruction.

16 But, exerting all my powers to call upon God to deliver me out of the power of this enemy which had seized upon me, and at the very moment when I was ready to sink into despair and abandon myself to destruction—not to an imaginary ruin, but to the power of some actual being from the unseen world, who had such marvelous power as I had never before felt in any being—just at this moment of great alarm, I saw a pillar of light exactly over my head, above the brightness of the sun, which descended gradually until it fell upon me.

17 It no sooner appeared than I found myself delivered from the enemy which held me bound. When the light rested upon me I saw two Personages, whose brightness and glory defy all description, standing above me in the air. One of them spake unto me, calling me by name and said, pointing to the other—This is My Beloved Son. Hear Him!

18 My object in going to inquire of the Lord was to know which of all the sects was right, that I might know which to join. No sooner, therefore, did I get possession of myself, so as to be able to speak, than I asked the Personages who stood above me in the light, which of all the sects was right (for at this time it had never entered into my heart that all were wrong)—and which I should join.

19 I was answered that I must join none of them, for they were all wrong; and the Personage who addressed me said that all their creeds were an abomination in his sight; that those professors were all corrupt; that: “they draw near to me with their lips, but their hearts are far from me, they teach for doctrines the commandments of men, having a form of godliness, but they deny the power thereof.”

20 He again forbade me to join with any of them; and many other things did he say unto me, which I cannot write at this time.

We don't know what else they discussed. Perhaps Joseph asked to see Christ's scars. Though, personally, I imagine he simply took them at their word.

Your points are valid enough, and someone with more experience may have asked for more authentication in light of the scriptures you've shared. Though there are plenty of scriptures depicting people seeing God, or promises of seeing God, and no mention of any of them asking for proof of identification.

In the year that King Uzziah died I saw the Lord sitting upon a throne, high and lifted up; and the train of his robe filled the temple. Isaiah 6:1

And I said: “Woe is me! For I am lost; for I am a man of unclean lips, and I dwell in the midst of a people of unclean lips; for my eyes have seen the King, the Lord of hosts!” - Isaiah 6:5

Beloved, we are God's children now, and what we will be has not yet appeared; but we know that when he appears we shall be like him, because we shall see him as he is. - 1 John 3:2

“Blessed are the pure in heart, for they shall see God." - Matthew 5:8

"And there has not arisen a prophet since in Israel like Moses, whom the Lord knew face to face," - Deuteronomy 34:10

So Jacob called the name of the place Peniel, saying, “For I have seen God face to face, and yet my life has been delivered.” - Genesis 32:30

Thus the Lord used to speak to Moses face to face, as a man speaks to his friend. - Exodus 33:10-11

And they saw the God of Israel. There was under his feet as it were a pavement of sapphire stone, like the very heaven for clearness. Exodus 24:10

But Stephen, full of the Holy Spirit, looked up to heaven and saw the glory of God, and Jesus standing at the right hand of God. “Look,” he said, “I see heaven open and the Son of Man standing at the right hand of God.” - Acts 7:55-56