Find equal columns between two dataframes

One way of merge

s=df1.T.reset_index().merge(df2.T.assign(match=lambda x : x.index))
dict(zip(s['index'],s['match']))
{'a1': 'b5', 'a2': 'b7', 'a3': 'b6', 'a4': 'b4', 'a5': 'b1', 'a6': 'b3', 'a7': 'b2'}

Here is a way using sort_values:

m=df1.T.sort_values(by=[*df1.index]).index
n=df2.T.sort_values(by=[*df2.index]).index
d=dict(zip(m,n))
print(d)

---

{'a1': 'b5', 'a5': 'b1', 'a2': 'b7', 'a3': 'b6', 'a6': 'b3', 'a7': 'b2', 'a4': 'b4'}

Here's one way leveraging broadcasting to check for equality between both dataframes and taking all on the result to check where all rows match. Then we can obtain indexing arrays for both dataframe's column names from the result of np.where (with @piR's contribution):

i, j = np.where((a.values[:,None] == b.values[:,:,None]).all(axis=0))
dict(zip(a.columns[j], b.columns[i]))
# {'a7': 'b2', 'a6': 'b3', 'a4': 'b4', 'a2': 'b7'}