Is shunning/disfellowshipping biblical in Protestant churches?

At least some churches, such as mine, which is protestant, practice what is called church discipline, which although slightly different is very similar to what you describe and is based on Matthew 18:15-17,

15 “If your brother sins against you, go and tell him his fault, between you and him
alone. If he listens to you, you have gained your brother. 16 But if he does not
listen, take one or two others along with you, that every charge may be established
by the evidence of two or three witnesses. 17 If he refuses to listen to them, tell
it to the church. And if he refuses to listen even to the church, let him be to you
as a Gentile and a tax collector.

Thus if someone is sinning significantly, for example having an affair (discretion is used when determining whether a particular sin is significant, as we are all sinners; you wouldn't get kicked out for getting a speeding ticket), they would be confronted by most likely a close friend, then one or two church elders, then the matter would be brought before the congregation and if they still refused to change their ways, they would be asked to leave the church - no time period is set though, and I imagine if they did eventually repent they might be welcomed back.

The general idea of not including unrepentant sinners is found in several places in the New Testament, such as in this passage and 1 Corinthians 5.

It is biblically supported:

It is reported commonly that there is fornication among you, and such fornication as is not so much as named among the Gentiles, that one should have his father's wife. ... Therefore put away from among yourselves that wicked person. (1 Cor. 5:1,13 KJV)

From a psychology perspective this is common social behaviour outside the church too—if someone commits egregious behavior and is unrepentant, then judgement (including ostracizing) can help to conform that person's behavior to something socially acceptable.

However

You also mention this:

I cannot personally accept it as being a right thing to do.

Your intuition isn't wrong here. To put someone outside the fellowship of the church is arguably the worst kind of suffering you can inflict on them, since you are affecting not their body but their soul and spirit, and their position in the Body of Christ. Consider these verses:

But now hath God set the members every one of them in the body, as it hath pleased him. (1 Cor. 12:28 KJV)

But whoso shall offend one of these little ones which believe in me, it were better for him that a millstone were hanged about his neck, and that he were drowned in the depth of the sea. (Matt. 18:6 KJV)

When Jesus had lifted up himself, and saw none but the woman [who had been caught committing adultery, the same sin as in the 1 Cor 5 example], he said unto her, Woman, where are those thine accusers? hath no man condemned thee? She said, No man, Lord. And Jesus said unto her, Neither do I condemn thee: go, and sin no more. (John 8:10-11 KJV, square brackets mine)

So my conclusion with all these verses is that yes, there is biblical basis for disfellowshipping, but I would very much hesitate to do so, lest I find myself:

• An enemy of the Body of Christ
• Stumbling one of God's own children
• Condemning someone Christ Himself does not condemn
• Placing myself in a position worse off than death-by-drowning.

The Bible makes it quite clear that one should avoid associating with a fellow Christian that continues in sin:

But now I have written unto you not to keep company, if any man that is called a brother be a fornicator, or covetous, or an idolater, or a railer, or a drunkard, or an extortioner; with such an one no not to eat.
— 1 Corinthians 5:11, KJV

It's not obvious however that someone in this state should be shunned for a fixed minimum period of time, which feels more like a form of punishment.
(If someone has demonstrably repented and changed their ways, why shouldn't they be forgiven and welcomed back?)

On the other hand, forgiveness doesn't necessarily mean forgetting. E.g. if someone's sin involved stealing from the collection plate, or acting inappropriately with children, they can be forgiven, but they should also be kept away from similar temptation. And in the case of theft, they should still be expected to repay the debt.

Apart from the Pentecostal church, shunning is practiced by the Amish, the Exclusive Brethren (Plymouth Brethren), Jehovah’s Witnesses and the Church of Scientology. However, Jehovah’s Witnesses say they are neither Protestant nor Catholic, and I doubt if Scientology is viewed as a Protestant denomination.

There are two issues here, shunning and disfellowshipping. It is possible to disfellowship (or excommunicate) someone from the church without shunning them. Indeed, the Bible gives grounds for putting the unrepentant wrongdoer out of the church.

I wrote to you in my letter not to associate with sexually immoral people — not at all meaning the people of this world who are immoral, or the greedy and swindlers, or idolaters. In that case you would have to leave this world. But now I am writing to you that you must not associate with anyone who claims to be a brother or sister but is sexually immoral or greedy, an idolater or slanderer, a drunkard or swindler. Do not even eat with such people. What business is it of mine to judge those outside the church? Are you not to judge those inside? God will judge those outside. “Expel the wicked person from among you” (1 Corinthians 5:9-13)

If your brother sins against you, go and rebuke him in private. If he listens to you, you have won your brother. But if he won’t listen, take one or two more with you, so that by the testimony of two or three witnesses every fact may be established. If he pays no attention to them, tell the church. But if he doesn’t pay attention even to the church, let him be like an unbeliever and a tax collector to you (Matthew 18:15-17).

Disfellowshipping is a term that refers to putting one out of the fellowship, or common group, due to some sin, moral lapse, or unfaithfulness. Such church discipline is to be removed from membership and should be applied to unrepentant sinners.

In Matthew 18 Jesus teaches that, if a person who claims to be a believer will not repent of specific sin after several confrontations, that person should be treated as an unbeliever. Paul also addresses church discipline: “I wrote to you in my letter not to associate with sexually immoral people—not at all meaning the people of this world who are immoral, or the greedy and swindlers, or idolaters. In that case you would have to leave this world. But now I am writing to you that you must not associate with anyone who claims to be a brother or sister but is sexually immoral or greedy, an idolater or slanderer, a drunkard or swindler. Do not even eat with such people. What business is it of mine to judge those outside the church? Are you not to judge those inside? God will judge those outside. Expel the wicked person from among you” (1 Corinthians 5:9–13). So, unrepentant sinners within the church are to be removed from the local body. However, Paul reminds the church in 2 Corinthians 2:7–8 to restore and forgive a repentant brother. The ultimate goal of excommunication or disfellowshipping is repentance and the restoration of fellowship. https://www.gotquestions.org/disfellowshipping.html

Shunning friends and family who have been disfellowshipped from a religious denomination is not biblical. Job was “blameless and upright, a man who fears God and shuns evil” (Job 1:8). Job confessed that “the fear of the Lord—that is wisdom, and to shun evil is understanding” (Job 28:28). The Bible advises us, “Do not be wise in your own eyes; fear the Lord and shun evil” (Proverbs 3:7–8). “A wise man fears the Lord and shuns evil” (Proverbs 14:16). Shunning evil is good, but to call a former associate “evil” because they disagreed with the teachings of the leadership and resigned or walked away is not biblical. It is a form of control and is the hallmark of a cult.

Scripturally, excluding a person from the church is preceded by admonition and counsel; it is only employed in cases of bona fide heresy, obdurate divisiveness, or blatant, unrepentant sin; and it is a last resort. After excommunication, the relationship between the former member and the church naturally changes, and the “shunning command”—not to eat with such a person—may come into play. However, the church still has the responsibility to pray for the one being disciplined and to extend forgiveness when repentance is evident. Shunning, as defined as a refusal to speak to someone or a total severing of all ties, goes beyond what the Bible advocates. https://www.gotquestions.org/Bible-shunning.html

Some churches twist Scripture to give more authority to the leadership and keep the members under their control. One example is the use of Hebrews 13:17 as a basis for demanding unquestioning loyalty and obedience to the leaders. Some religious groups view questioning the leaders as tantamount to questioning God. Some leaders claim to have divine authority and approval; thus, to disobey them is to disobey God. This is perhaps the most pernicious form of spiritual manipulation, and it has no place in a true church.

Victims of spiritual manipulation seldom realize what’s happening to them. Here are some indicators of a spiritually manipulative church:

Legalism

Demands for obedience

Unquestioning submission

Punishment (loss of privileges, shunning, or expulsion)

Misplaced loyalty

Emphasis on performance

Exclusivism (“we alone are right, and everybody else is wrong”)

Isolation (refusal to associate with anyone but spiritual brothers and sisters)

Humiliation of the "disobedient"

https://www.gotquestions.org/spiritual-manipulation.html

More related information here: https://www.gotquestions.org/spiritual-abuse.html

To conclude: It is biblical to put an unrepentant sinner out of the church. It is unbiblical to shun family members and former friends simply because the leadership of the church says you must. Let the Bible have the final word:

Do not turn away from your own flesh and blood (Isaiah 58:7).