@dannyeuu's answer is correct. pd.concat naturally does a join on index columns, if you set the axis option to 1. The default is an outer join, but you can specify inner join too. Here is an example:
x = pd.DataFrame({'a': [2,4,3,4,5,2,3,4,2,5], 'b':[2,3,4,1,6,6,5,2,4,2], 'val': [1,4,4,3,6,4,3,6,5,7], 'val2': [2,4,1,6,4,2,8,6,3,9]})
x.set_index(['a','b'], inplace=True)
x.sort_index(inplace=True)
y = x.__deepcopy__()
y.loc[(14,14),:] = [3,1]
y['other']=range(0,11)
y.sort_values('val', inplace=True)
z = x.__deepcopy__()
z.loc[(15,15),:] = [3,4]
z['another']=range(0,22,2)
z.sort_values('val2',inplace=True)
pd.concat([x,y,z],axis=1)