Is it permissible in Sharia to breastfeed a five-year-old girl in order to establish a kinship tie, and if one's sister breastfeeds, does that make her an aunt to the suckling, and does the child thereby become unmarriageable to the nursing mother and her daughter through suckling?

There is no harm in sponsoring a girl, and reward is hoped for from Allah, the Almighty. If the breastfeeding occurs within the first two years (hawlain), it establishes a prohibited degree of kinship (hurmah). Thus, the girl breastfed by your sister becomes your sister's daughter through suckling. If you breastfeed the boy you wish to sponsor, he becomes your son through suckling, and your sister's son, meaning you become his maternal aunt through suckling. This makes him a mahram (definition) to you and a brother to the girl through suckling. This is the preferred approach if you intend to sponsor him, so that you become a mahram to him. The same applies to the girl under your sponsorship. In this case, there would be no harm if he reaches puberty in his entering upon you or the girl, or in being in khalwah with him, and other things permissible for a woman with her mahrams. However, we emphasize that the breastfeeding must occur within the first two years. As for breastfeeding an adult, it has no effect.