How do you know if someone is linked with another account in the forum?
There are multiple ways for this, The most likely one is that you use the same address on both accounts. Contact details, Country, Ip and so on.
What is the basis for branding such accusation?
Mostly you will get in trouble when you exploit Signature / Airdrops / Bounty Campaigns. Others may include, Giving a loan to your alt accounts and using to boost their trust. The list is pretty big one.
If accused, what can we do about it?
If it is proven that both email/btc addresses are of the same user and he is exploiting a campaign then there isn't much you can do about that. Unless a user is blaming you without any connection, your chances of getting free from the red trust are very low.
How do we make an appeal? To whom?
You can't, Trust isn't moderated. Anyone can leave a feedback. You can only appeal to the person who is accusing you.
There are many users of this forum and coincidence cannot be prevented when having the same date and time of registration and entering the same bounty campaign.
This alone can't be the reason for the red trust imo.