Their family background could be really diverse. Maybe they have a family full of strong - willed witches. Their parents could be part of an old wizarding family that has been somewhat on the outskirts of the mainstream magical society. This would explain Hermione's thirst for knowledge as she tries to bring her family more into the limelight in the magical world, while Tonks' more rebellious nature could also stem from this family position.
Perhaps their family was a blend of different magical cultures. One part of the family could have come from a more European wizarding tradition where Hermione's studious nature was nurtured, and another part could have been from a more wild and free - spirited magical group where Tonks' shapeshifting and carefree attitude originated. Their family might have had a hidden library or collection of magical artifacts that Hermione was always eager to explore, and Tonks was more interested in the stories and legends behind them.
Well, it could be that their family has a long history of shapeshifting abilities which Tonks inherited more obviously, and Hermione got more of the intellectual and academic magical abilities. Their family might have been persecuted in the past for some reason, perhaps being suspected of dark magic because of their unique abilities. This would make both Hermione and Tonks more determined to prove themselves in the wizarding world, each in their own way.