First flatten values by numpy.ravel and reshape by original df, use DataFrame constructor and create columns names, last join to original:
df1 = pd.DataFrame(pd.factorize(df.values.ravel())[0].reshape(df.shape))
df1.columns = ['ID{}'.format(x+1) for x in range(len(df1.columns))]
print (df1)
ID1 ID2
0 0 1
1 0 2
2 1 3
3 2 4
4 3 4
df = df.join(df1)
print (df)
Name1 Name2 ID1 ID2
0 John Jack 0 1
1 John Albert 0 2
2 Jack Eva 1 3
3 Albert Sara 2 4
4 Eva Sara 3 4
Create MultiIndex Series by stack, create ids by factorize and for DataFrame unstack, then rename columns and add to original by join:
s = df.stack()
df = df.join(pd.Series(pd.factorize(s)[0], index=s.index)
.unstack()
.rename(columns=lambda x: x.replace('Name','ID')))
print (df)
Name1 Name2 ID1 ID2
0 John Jack 0 1
1 John Albert 0 2
2 Jack Eva 1 3
3 Albert Sara 2 4
4 Eva Sara 3 4
Similar alternative:
s = df.stack()
s[:] = pd.factorize(s)[0]
df = df.join(s.unstack().rename(columns=lambda x: x.replace('Name','ID')))
print (df)
Name1 Name2 ID1 ID2
0 John Jack 0 1
1 John Albert 0 2
2 Jack Eva 1 3
3 Albert Sara 2 4
4 Eva Sara 3 4