Convert List of Associations into Association of Lists

You can use the undocumented AllowedHeads option of Transpose to do this:

tr = Transpose[listData, AllowedHeads -> {Association, List}]

<|"Input" -> {1, 2, 3}, "double" -> {2, 4, 6}, "squared" -> {1, 4, 9}|>

and back:

Transpose[tr, AllowedHeads -> {Association, List}]

{<|"Input" -> 1, "double" -> 2, "squared" -> 1|>, <|"Input" -> 2, "double" -> 4, "squared" -> 4|>, <|"Input" -> 3, "double" -> 6, "squared" -> 9|>}

Merge[listData, Identity]

<|"Input" -> {1, 2, 3}, "double" -> {2, 4, 6}, "squared" -> {1, 4, 9}|>

listFormatData = {<|"Input" -> input1,  "OutputKey1" -> OutputKey1Input1Output|>,
   <|"Input" -> input2, "OutputKey1" -> OutputKey1Input2Output|>};
Merge[listFormatData , Identity]

<|"Input" -> {input1, input2}, "OutputKey1" -> {OutputKey1Input1Output, OutputKey1Input2Output}|>

A function that is of great use here, is AssociationTranspose in the GeneralUtilities package. It does the same thing as regular transpose, but works with List and Associations alike (and mixtures of the two):

listData = {
  <|"Input" -> 1, "double" -> 2, "squared" -> 1|>, 
  <|"Input" -> 2, "double" -> 4, "squared" -> 4|>,
  <|"Input" -> 3, "double" -> 6, "squared" -> 9|>
};
assocData = <|"Input" -> {1, 2, 3}, "double" -> {2, 4, 6}, "squared" -> {1, 4, 9}|>;
GeneralUtilities`AssociationTranspose[listData]  === assocData
GeneralUtilities`AssociationTranspose[assocData] === listData

Out[11]= True

Out[12]= True