As a big fan of the sibling relationship Silver and Blue have going on in pokespe, in my version of the gameverse they still grew up together prior to Giovanni's departure.
[CLARIFICATION TO WHO I MEAN WHEN I REFER TO BLUE OR GREEN: tl;dr I generally refer to the Kanto rival as Green, and the girl we see in Pokespe and LGPE as Blue]
Blue's parents were active members of Team Rocket since before she was born (probably from one of the older waves of recruits from Giovanni's early efforts to widen the scope of Rocket after he came into power), and she spent a lot of time growing up bouncing around the bases. Rocket had been around in one form or another for a couple generations, even if it's had its ups and downs, a few re-brands, and waves of new blood over the decades (I'd place Giovanni in his early 40s during the events of RBY, and his mother was probably a Rocket lifer, herself, though it wasn't after until it came to Giovanni's hands in the 80s that it was called Team Rocket), and there's a number of people of various ages who were born and raised into this.
Blue and Silver were seen as the newest generation of Rocket lifers and met when they were both little, after Silver was born and was old enough to be shoved off to babysitters (I quite like Ariana and have some very specific thoughts on her, but for now I'll just say she never wanted to be a mother, and as soon as she could say "you figure out what to do with the baby, Giovanni, I've done my part," she did, and returned to her rewarding life of crime full time without looking back once).
So they ended up spending a lot of time together because they were both kids in the base having their care outsourced to a mix of the most excellent babysitters Giovanni's crooked money could buy, and also the likes of Petrel (who got along well enough with children while being a fabulously bad influence). Blue, being 3 years older than Silver, just sort of naturally assumed the position of Silver's big sister. On the adults' side, there was some sort of undercurrent of understanding that one day, when Silver inherited Giovanni's position, it would benefit Silver to have at least one executive who would be totally loyal and trustworthy, and the best way to ensure that might be to allow this sort of bond to form between them.
Of course, after Blue left on her own unofficial version of a pokemon journey during the events of RBY, they end up not seeing each other for far longer than intended. Blue's been been gone for a few months when things go down with Red and Giovanni, and by the time Blue returns, Silver has already had her own confrontation with Giovanni abandoning her, and has left for destinations unknown. 3 years later, when the GSC protagonist encounters Silver at Mt. Moon, she has her own reasons for being in Kanto. In part, having grown past the ideals she was raised with, gotten some closure for her abandonment, and started down a path to growing into a new person, she was ready to seek out her big sister and share who she's becoming.
