Of course not: you are mistaking sum types and product types.
Even if we assume they are product types, [A×B×C] ≃ [A]×[B]×[C] is still not correct. The former doesn't allow the number of As and Bs and Cs to differ: the latter does. So the latter strictly speaking admits more values than the former.
Since both of them allow infinite number of values, to make a size comparison we will use generating functions. A list of A (or [A] in your notation) allows an empty list (one possible value), a one-element list (as many values as A itself), a two-element list… which becomes 1+A+A^2+…=1/(1-A). The beautiful thing about calling them sum types or products types is that you can manipulate it just by summing or multiplying respectively.
So the number of values for [AxBxC] is identified by 1/(1-ABC). For [A]*[B]*[C] it's 1/(1-A)/(1-B)/(1-B) which simplifies to 1/(1-A-B-C+AB+AC+BC-ABC). Now it becomes obvious† this form admits more values and you can in a sense quantify how many more!
†: Okay perhaps it's only obvious if I also include a Venn diagram but diagramming is beyond what I can do on HN.
Yeah, and if you carefully expand 1+(A+B+C)+(A+B+C)^2+(A+B+C)^3+..., there'll be strictly more summands than in (1+A+A^2+A^3+...)×(1+B+B^2+B^3+...)×(1+C+C^2+C^3+...) due to more flexible ordering of elements, so they're not isomorphic.
I still think it's the most obvious thing that [A+B+C] at least maps surjectively onto [A]×[B]×[C] :)
Even if we assume they are product types, [A×B×C] ≃ [A]×[B]×[C] is still not correct. The former doesn't allow the number of As and Bs and Cs to differ: the latter does. So the latter strictly speaking admits more values than the former.
Since both of them allow infinite number of values, to make a size comparison we will use generating functions. A list of A (or [A] in your notation) allows an empty list (one possible value), a one-element list (as many values as A itself), a two-element list… which becomes 1+A+A^2+…=1/(1-A). The beautiful thing about calling them sum types or products types is that you can manipulate it just by summing or multiplying respectively.
So the number of values for [AxBxC] is identified by 1/(1-ABC). For [A]*[B]*[C] it's 1/(1-A)/(1-B)/(1-B) which simplifies to 1/(1-A-B-C+AB+AC+BC-ABC). Now it becomes obvious† this form admits more values and you can in a sense quantify how many more!
†: Okay perhaps it's only obvious if I also include a Venn diagram but diagramming is beyond what I can do on HN.