As stated in merge, join, and concat documentation, ignore index will remove all name references and use a range (0…n-1) instead. So it should give you the result you want once you remove ignore_index argument or set it to false (default).
df = pd.concat([df, df2], axis=1)
This will join your df and df2 based on indexes (same indexed rows will be concatenated, if other dataframe has no member of that index it will be concatenated as nan).
If you have different indexing on your dataframes, and want to concatenate it this way. You can either create a temporary index and join on that, or set the new dataframe’s columns after using concat(…, ignore_index=True).