Pretty sure you cant have ws defined as a worksheet and assign a string to it.
If you change Dim ws as worksheet to Dim ws as variant it works
Code:
Sub Einfügen_New_A_and_Fill_with_Content_SPECIFIC_SHEET()
Dim ws As Variant, Data$
ws = InputBox("Provide a sheet name:")
Data = InputBox("What would you like to insert in column A ?")
For Each ws In Sheets
ws.Columns(1).Insert
ws.Cells(1).Resize(ws.UsedRange.Rows.Count) = Data
Next
End Sub