Copy Column Based On Its Label

I am trying to figure if my question is possible and if so, how to do it. I am trying to copy and entire excel column into another sheet based on the label of that column. For example: Across Row 1 would be the labels "DOG" or "CAT", depending on the data below. When a column is labeled "DOG" I want the data in that column to be copied into the "DOG" sheet and have the same happen for "CAT". If you need any further clarification please let me know. Any help is greatly appreciated.

Can you give more specific examples with a spreadsheet? You could do some type of lookup function where you put the name of the sheet in a cell hidden somewhere on each tab and then lookup the data that way. If you want to refer to the sheet, you'd have to use VBA - I don't know a way to reference a sheet name in an excel formula per se.

pet example.xlsx

Here's an example of how I would want the spreadsheet to look. I had to manually input the DOG an CAT data into the "Dogs" and "Cats" spreadsheets whereas I would like it to be done automatically.

I'm not really sure what you mean with your example, but the file I uploaded is exactly how I would want the final format to look.