What is the biblical basis for the belief that God likes or feels affection toward us?

The best answer to your question was given in the answer by Michael Hoffman. That being John 3:16 Why would God sacrifice his only son to save us from the eternal torture unless he loved us as much as he loves Jesus. We are created in the image of God, that does not mean the physical, but the Spiritual image;

Genesis 1:27 KJV So God created man in his own image, in the image of God created he him; male and female created he them.

Jesus himself told us that God is Spirit.

John 4:24 KJV God is a Spirit: and they that worship him must worship him in spirit and in truth.

Therefore, if our Spirit is in the Image of the Spirit God then our love for our children must somewhat mirror God's love for his son.

But there are also other Scriptures which show God's love for all of his creation, of which we are a part. but God's special love for Mankind is spread throughout the Scriptures. God gave man dominion over all the animals;

Genesis 1:28 KJV And God blessed them, and God said unto them, Be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth.

I could quote Scripture after Scripture showing God's love for man, but Jesus himself said it best when he said;

Matthew 18:14 KJV Even so it is not the will of your Father which is in heaven, that one of these little ones should perish.

On the surface this may not seem to be much of a statement until we understand that Jesus knew all of the circumstance of his death as he said this. He knew at that time the pain and suffering ahead of him which he did not deserve. We must also take into consideration that he also knew that many of those for whom he would sacrifice would reject him and even denigrate his sacrifice. We also need to know that God the father was aware of these things, yet he would sacrifice his son anyway. Jesus also gave his life willingly for us:

Matthew 26:53 KJV Thinkest thou that I cannot now pray to my Father, and he shall presently give me more than twelve legions of angels?

Finally God Not only loves , but God is love.

1st John 4:16 and 17 KJV And we have known and believed the love that God hath to us. God is love; and he that dwelleth in love dwelleth in God, and God in him. 17 Herein is our love made perfect, that we may have boldness in the day of judgment: because as he is, so are we in this world.

Should you need more Scriptures which show the Love of God I suggest you read the Holy Bible from Genesis through Revelation, and underline those that you find. Be forewarned that you are going to need several pens.

Is there biblical basis for the belief that God's love for people includes a genuine individual affection?

In Luke 7:36-50, we have some valuable insight into the nature of the phenomenon known to us as, "love".

Jesus relates a parable about a creditor who forgives two debtors what they owe. He then asks of Simon, the pharisee who had invited him to the meal, "Which of them [the debtors] will love him [the creditor] more?". Simon speculates that it would be the one who was forgiven the most, an answer which Jesus commends.

Both the pharisee and Jesus expected the debtors to return something to the creditor in exchange for the debt that was forgiven -- LOVE. There is no evidence in the parable that the creditor got what he paid for, but five hundred and fifty denarii was paid for it, nevertheless. I'm pretty sure most people, though, would concur with the pharisee and Jesus that love was a real expectation, and that it wasn't beyond the means of either debtor to render it.

When/if the debtors were to render the creditor his due, what would have been the trigger for it? The answer, to me, is obvious: their love for him would be born of the love that had been first shown to them -- the creditor "procured" the love of the debtors by forgiving them their debts.

John says it quite plainly:

We love Him because He first loved us.
-- 1 John 4:19

In regard to lending, Jesus says this:

And if you lend to those from whom you hope to receive back, what credit is that to you? For even sinners lend to sinners to receive as much back. But love your enemies, do good, and lend, hoping for nothing in return; and your reward will be great, and you will be sons of the Most High. For He is kind to the unthankful and evil.
-- Luke 6:34-35

We are compelled by Jesus to lend, not with no expectation of return, but with no expectation of receiving back what was given. We can quite legitimately expect our generosity to procure for us a reward: firstly, the promised love of the Father -- "your reward will be great"; and secondly, the love of those to whom we lend.

The Bible is replete with examples of God's love for individuals, starting at the very beginning with Adam:

So Adam gave names to all cattle, to the birds of the air, and to every beast of the field. But for Adam there was not found a helper comparable to him. And the LORD God caused a deep sleep to fall on Adam, and he slept; and He took one of his ribs, and closed up the flesh in its place. Then the rib which the LORD God had taken from man He made into a woman, and He brought her to the man.

God endeavored to procure Adam's love by making him a wife. Was God successful? Well, Adam would have to answer for himself. It would be a great exegetical exercise, though, to explore whether He was or wasn't, and also to give consideration to what it cost God to meet Adam's need.

Then there was Cain and Noah and Abraham, etc, etc. Wherever it can be seen in scripture that God has met the needs of his people, one has evidence of God's genuine individual affection. You won't see it, however, if you don't feel it yourself, before you start looking.

Conclusion

The question the OP posed can only really be answered from a personal perspective, i.e. "Is God's love for me genuine?" If people feel loved by God then they will have no problem rendering to Him His due, and also be able to clearly articulate what it cost Him to procured it.

The New Testament states God is love (ἀγάπη):

Anyone who does not love does not know God, because God is love. (1 John 4:8 ESV)

So we have come to know and to believe the love that God has for us. God is love, and whoever abides in love abides in God, and God abides in him. (1 John 4:16 ESV)

The New Testament gives this definition of love (ἀγάπη):

In this is love, not that we have loved God but that he loved us and sent his Son to be the propitiation for our sins. (1 John 4:10 ESV)
No one has ever seen God; if we love one another, God abides in us and his love is perfected in us. (1 John 4:12 ESV)

God's love (ἀγάπη) is defined by His actions:

• God loved us
• God sent His Son to be a propitiation for our sins
• God's love is perfected in us

His affection for people is not only stated, it is demonstrated by His actions, which are continuous and on-going: "God's love is perfected in us."

I would say that Luke 15:7 is another that supports both truths, that God is affectionate towards us and that He cares about us individually: “I will tell you that in the same way there will be more rejoicing in heaven over one sinner who repents than over ninety-nine righteous persons who do not need to repent.” (NIV)

I know it’s been a while since this was posted, but thank you for raising the question and for everyone’s thoughtful responses. I needed this reminder today.

I believe we know from the epistles that God feels affection for individuals. Since the epistle authors were writing under the inspiration of the Holy Spirit, their attitudes and words should correctly convey God’s emotions towards us.

Philippians 1:8

God can testify how I long for all of you with the affection of Christ Jesus.

Paul even attributes his emotions directed at the members of this church to Jesus Christ Himself. He isn’t filled with Paul’s affection - it’s Jesus’ affection filling Paul.

https://www.google.com/amp/s/www.biblestudytools.com/blogs/matthew-s-harmon/philippians-the-affection-of-christ-jesus.html%3famp

Also, God relates to us as children. What healthy father doesn’t feel affection for His individual child?

God refers to Abraham as his friend. Individually. Isaiah 41:8

But you, Israel, are My servant, Jacob whom I have chosen, The descendants of Abraham My friend.

God grieved over Saul.

1 Samuel 15:11

"I regret that I have made Saul king, because he has turned away from me and has not carried out my instructions.” Samuel was angry, and he cried out to the LORD all that night.

He personally dealt with David - David’s personal relationship with the Lord covered every emotion imaginable, from delight to severe displeasure.

Psalm 18:19

He brought me out into a spacious place; he rescued me because he delighted in me.

2 Samuel 12:1-10

But the thing that David had done displeased the Lord, and the Lord sent Nathan...

God is presented in scripture as king. We are presented as bride. The very purpose, the primary goal of the bride (apart from producing offspring) is to engage intimately with the king.

Yes, I believe God does have genuine affection for us individually.

Remember, the word affection has to do with affect, or emotion. To be affected by someone is to have one's emotions stirred in some way, be it in a positive or negative way. Besides its use as a verb, the word affect can also be used as a noun:

His affect was remarkably subdued.

To psychologists, affect denotes a "subject's externally displayed mood."

We need not search the Gospels (or the epistles, for that matter) too hard to observe how Jesus was affected by individuals, groups of people, and even cities and situations. Here is a brief list (with my bold print):

• "O Jerusalem, Jerusalem, the city that kills the prophets and stones those sent to her! How often I wanted to gather your children together, just as a hen gathers her brood under her wings, and you would not have it!" (Luke 13:34 NASB). Notice the motherly affection and care Jesus had for the city where he would soon be crucified. Notice, too, how he was moved with compassion here despite the shameful history of Jerusalem regarding their treatment of God's prophets--including Jesus!

• Then, looking at him, Jesus loved him and said to him, "You lack one thing: Go, sell all you have and give to the poor, and you will have treasure in heaven. Then come, follow Me" (Luke 10:21).

• Peter, turning around, saw the disciple whom Jesus loved following them; the one who also had leaned back on His bosom at the supper and said, "Lord , who is the one who betrays You?" (John 21:20)

• When Mary reached the place where Jesus was and saw him, she fell at his feet and said, "Lord, if you had been here, my brother would not have died." When Jesus saw her weeping, and the Jews who had come along with her also weeping, he was deeply moved in spirit and troubled. "Where have you laid him?" he asked. "Come and see, Lord," they replied. Jesus wept. Then the Jews said, "See how he loved him!" But some of them said, "Could not he who opened the eyes of the blind man have kept this man from dying?"

• Moved with compassion, Jesus touched their eyes; and immediately they regained their sight and followed Him (Matthew 20:34 NAS).

• Moved with compassion, Jesus stretched out His hand and touched him, and said to him, "I am willing; be cleansed" (Mark 1:41 NAS).

• For we do not have a high priest who cannot sympathize with our weaknesses, but One who has been tempted in all things as we are, yet without sin (Hebrews 4:15).

• Are not two sparrows sold for a penny? and not one of them shall fall on the ground without your Father: but the very hairs of your head are all numbered. Fear not therefore: ye are of more value than many sparrows (Matthew 10:29-31 ASV).

I may add to this list at some point, but I think the list as it is gives us at least a clue as to how God feels about us, empathizes with us, and cares about us. Let me conclude with some verses from the Psalms:

 Many, O Jehovah my God, are the wonderful works which thou hast done, 

 And thy thoughts which are to us-ward; 

 They cannot be set in order unto thee; 

 If I would declare and speak of them, 

 They are more than can be numbered (Psalm 40:5 ASV). 

And,

 You surround me—front and back. 

 **You put your hand on me**.

 That kind of knowledge is too much for me; 

 It's so high above me that I can't fathom it. 

 You are the one who created my innermost parts; 

 You knit me together while I was still in my mother's womb.

 I give thanks to you that I was marvelously set apart. 

 Your works are wonderful—I know that very well.

 My bones weren't hidden from you when I was being put together 

 In a secret place, when I was being woven together 

 In the deep parts of the earth.

 Your eyes saw my embryo, 

 And on your scroll every day was written 

 That was being formed for me, 

 Before any one of them had yet happened (Psalm 139, passim).     

John 3:16:

For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.

Romans 8:16-17:

The Spirit itself beareth witness with our spirit, that we are the children of God: And if children, then heirs; heirs of God, and joint-heirs with Christ; if so be that we suffer with him, that we may be also glorified together. For I reckon that the sufferings of this present time are not worthy to be compared with the glory which shall be revealed in us.

There are also many, many instances of angelic messengers and servants of God being sent to serve, protect, and teach people, identified as individuals, and not just as groups or important leaders. See Genesis 31; 1 Kings 17; and Daniel 3 for a few.

Aside from the examples above, the New Testament is also replete with examples of God's love for us as His children. The recorded miracles performed by Jesus Christ were almost always done for individuals of apparently little or no great fame or reputation. The Messiah, sent to save all mankind, frequently spent His precious, limited time ministering to individuals.

Your question is specifically about the Bible, but I'd also like to add that the Book of Mormon has many plain, beautiful examples and explanations of the love of God. 2 Nephi 2-4, Enos 1, and Mosiah 27 contain a few of my favorites.