They might face the challenge of not being recognized by their past selves or other characters. Also, they could accidentally change things in a way that creates new and more dangerous villains. Another challenge could be dealing with the technology and power levels of the past, which are much lower than what they are used to.
It could be someone like Frieza. Frieza is extremely powerful and evil in the Dragon Ball universe. If he somehow found his way into the Naruto universe or if the fusion character was in the Dragon Ball universe, he would be a great threat. His ability to transform and his cold - blooded nature would make him a tough opponent for the fusion of Goku and Naruto.
Goku's powers are on a much higher scale. His ability to destroy planets as a Super Saiyan makes him far more powerful than Naruto at his strongest. Naruto's jutsu are mainly focused on the ninja world, while Goku's powers are universal.
Often, in 'Goku Harem Fanfic', Goku becomes more aware of his romantic side. He might be a bit more in tune with the emotions of the female characters around him compared to his original portrayal in the Dragon Ball series, where he was mostly focused on fighting and training.
His anger often manifests physically. His muscles might bulge more than usual, and his hair could stand on end even more in his Super Saiyan forms. His ki, or energy, also becomes more visible and intense. It's like a red - hot aura surrounding him.
Loki would also be a difficult adversary. He's very cunning and could use his tricks and illusions to try and outwit Goku. Goku would need to rely on his senses and ki to see through Loki's deceptions.
His social skills might be different. Instead of the somewhat naive Goku we know, he could be more aware of social norms. Bulma could teach him about proper behavior, but at the same time, his Saiyan impulsiveness would still shine through at times. So he might be a strange mix of a well - mannered person in some situations and a wild fighter when provoked.
Vegeta could be a great rival. He might see Goku's new position as God of Destruction as another thing Goku has achieved that he hasn't. This could fuel their competitive nature even more.
His character would develop a more strategic side. Beerus is a very strategic fighter, and Goku, being raised by him, would learn to think more before acting in battles. He would also become more aware of the power hierarchy in the universe at a younger age, which could make him more cautious or more ambitious, depending on how you look at it. Additionally, his Saiyan pride might be tempered by Beerus' teachings, making him more of a team player in some ways.
Well, Vegeta would be a likely character. He might be initially suspicious of the female Goku. Maybe he thinks she's a threat or just can't believe there's another Goku - like being. He could be there to add some comical moments as he tries to understand the situation.