Is the definition of "sin" subjective? Of whom's morals is it based on?

What is and is not a sin is not subjective. Scripture gives a very clear definition of sin in 1 John 3:4.

Sin is transgression of the Law.

What is subjective is our understanding of what sin is. We're flawed finite creatures, with varying preconceptions, who are taught various different things (whether those things are right or wrong) and unfortunately we tend to filter everything we take in and understand it after it's been filtered by (or mixed with) our preconceptions.

So we disagree on certain points, That doesn't mean that Truth is subjective at all. There is still one Truth. Something is either a sin or it is not. We tend to disagree on certain things but the fact remains that it is a perfect, just, and holy God, not us, that determine what is a sin and what is not.

If the Old and New testaments were really giving differing definitions of sin, that would be a discrepancy. Since Scripture cannot contradict itself, we need to go to the tried-and-true rules for resolving apparent discrepancies.

In this case, it seems that your understanding of the idea of the New Testament promoting a "Subjective" view on what sin is seems to be a simple (and common) misunderstanding of historical context.

Your "Subjective" example seems to be based on Romans 14:14.

I know, and am persuaded by the Lord Jesus, that there is nothing unclean of itself: but to him that esteemeth any thing to be unclean, to him it is unclean.

What Paul is referring to is a concept known as "legalism" in which people try to say that you must obey the letter of the law to retain your salvation. This was a misuse of that Law. In effect, these people were saying that you are saved by obedience to the Law, rather than by the forgiveness of God through the sacrificial work of Christ.

In context, Paul is not saying that there is nothing that is inherently sinful. He's talking about people trying to enforce their own perspective of what is sinful on others. What he's saying is in perfect agreement with what Jesus said in Matthew 7:5

You hypocrite, first cast out the beam out of your own eye; and then shall you see clearly to cast out the speck out of your brother's eye.

_{Of course this doesn't mean that we should remain silent about sin. There is a time and a place, and a purpose for which the law exists, and we are commanded to rebuke sin. But only in the right time, for the right purpose, and not with a self-righteous attitude. More here.}

Put simply, it is a misconception that the New Testament teaches that sin is subjective. The New Testament as a whole agrees completely with the Old Testament. Sin is sin. But what the new Testament adds is that Christ offers freedom from sin.

There is a pretty good study on the issue at https://www.christiancourier.com/articles/1335-a-study-of-romans-14

Sin is not an idea or a component of a "moral system". God's laws are not based on a notion of right or wrong, but are based on God's personal nature and therefore how He wants us to be.

This doesn't mean that it's not wrong to sin - you can say to a degree that it invariably is wrong to sin - but to stress that sin itself isn't about good/bad, right/wrong, etc.

Sin is to deviate from this nature, but in the end God understands that we are not of His nature (which why we are said to be bound to sin, and that it is within our skin). This is why God is willing to show mercy on our sin and forgive us for not following the law. This is why He wanted to send his Son to accomplish this - because he knew sin is an actual part of what we are, making us dependent of a Savior.

Since I brought salvation into it I think I need to add that that a huge part of what God wants is for us to humble ourselves, specifically because of the ways we are not like Him and are dependent on his grace to be saved.

Sin is absolutely objective but in its application in our lives has a subjective element. The particular case that you mention is where Paul is saying if a person thinks something is a sin (even though it is not) then it is a sin to that person if they run about doing what their conscience condemns. This is because anything done without faith is a sin:

But whoever has doubts is condemned if he eats, because the eating is not from faith. For whatever does not proceed from faith is sin. (Romans 14:23, ESV)

In other words if you truly thought that wearing a purple shirt was a sin, you should not wear a purple shirt.

However the reverse is not true. Anything that is objectively a sin, like blasphemy, murder, sexual perversion, etc. even if you think it is not a sin, is still a sin. Sin is anything that is done not derived from pure love to God. This means we all sin daily as none perfectly loves God.

To answer your question: sin is objective (it is not loving God perfectly) but one objective sin is doing things you incorrectly consider as sin due to want of knowledge or wisdom. This is a sin because it shows lack of respect to God when you ignore your own conscience, even when it is confused on neutral matters.