Maybe but how about the cooking oil? - they were cooking together, dropped the oil and the floor was full of cooking oil?? Then they both fell to the floor and were swimming there, as in figure of speech (because the floor was literally full of oil)???
Actually, the explanation appears to be that mình does double duty as a reflexive pronoun (which was covered in the early lessons) and a first-person pronoun (an alternative to tôi used, they say, in friendly communication).
I don't think using ở would make sense. Trong is in and ở is at. You can't say "swimming in cooking oil" if ở is used. It would roughly translate to "swimming at cooking oil", which does not read smoothly in Vietnamese.