Rav Moshe Shternbuch: Do Adopted Children Sit Shiva For Their Step Parents?
Although they are technically not related or considered family, Rav Moshe Shternbuch says (3:374) that if an adopted child grew up with his step parents and considered them to be his parents he should sit Shiva and keep all halachos of Aveilus upon their death. While in Aveilus there is a concept of showing your sadness by keeping some rules of aveilus, Rav Shternbuch says that in this case one should keep all the details of aveilus exactly like an Aveil.
He learns this from Rabban Gamliel in the Mishna who sat Shiva for his slave, Tavi. The Rashba says that Rabban Gamliel sat shiva because he felt towards Tavi like a father would towards his son. Rav Shternbuch reasons that if feelings of closeness caused Rabban Gamliel to sit shiva for an Eved who has no Din of Aveilus than certainly for the people who raised you and do have a Din of Aveilus.
He adds that this is a Mitzva not a Chiyuv. Therefore you should not be Meikil when it comes to the Din of Aninus of not putting on Tefilin or excusing you before the Kevurah of any other mitzva. Lastly he says if you don't feel like a son to your step parents then you should just keep some halachos of Aveilus to share the pain but do not keep all the halachos like and aveil would.
Important Note: We try to convey the Tshuva to the best of our ability. We admit that our understanding may not be accurate. Please also understand that this Tshuva may not be the final word on this topic. One should consult a Rav before drawing any conclusions.