It could be Sirius Black. Sirius was a very powerful wizard and he had a special bond with Harry. He could have come back in some way to train Harry in some unique magical skills, like how to transform into a more powerful animagus form.
Well, perhaps in this fanfic, Harry Potter could be receiving special training in advanced defensive magic from a reclusive and powerful wizard. This training might include learning spells that are not taught at Hogwarts, like some ancient protection charms.
It could be Dumbledore. He's always been a guiding figure in Harry's life, so in the fanfic, he might continue to mentor Harry through his special training, teaching him ancient and powerful magic.
There's a possibility that it's a mysterious figure from a different wizarding school. This person could have heard of Harry's reputation and journey. They might give Harry the trivet as a sign of alliance or as a test. They want to see how Harry will use this strange object. Maybe this figure is from a school that has a different approach to magic, and the trivet is a symbol of that different magical culture.
Harry would become more disciplined. His natural leadership skills would be enhanced by the military training, making him even more of a force to be reckoned with.
One interesting plot could be the main character facing a series of extremely difficult and dangerous dark magic challenges during the auror training. Maybe they have to deal with a powerful dark wizard who has infiltrated the training grounds to disrupt the training process.
In some fanfics, Harry might go through combat training with new spells and dueling techniques. This could include advanced defensive magic like creating more complex Patronuses.
The trivet in the 'harry potter receives a trivet fanfiction' could be a central part of a new mystery. Maybe it has inscriptions on it that lead Harry on a quest to find out more about it. As he tries to decipher the meaning of these inscriptions, he meets new characters, both friendly and unfriendly. The trivet could also be a source of conflict, as other wizards might want to get their hands on it for their own purposes, and Harry has to protect it while also trying to figure out its true nature.
Well, it could be that Harry Potter gets a magical item from the Trival in the fanfiction. Maybe it gives him some new powers that he uses to fight against Voldemort in a different way.