BUG: DataFrame's behavior of boolean operators doesn't match pandas

[dchigarev]

Modin version checks

• I have checked that this issue has not already been reported.

• I have confirmed this bug exists on the latest released version of Modin.

• I have confirmed this bug exists on the main branch of Modin. (In order to do this you can follow this guide.)

Reproducible Example

from modin.pandas.test.utils import create_test_dfs

md_df1, pd_df1 = create_test_dfs({"a": [1, 2], "b": [3, 4]})
md_df2, pd_df2 = create_test_dfs({"b": [5, 2], "c": [6, 7]})

print(f"modin.eq:\n{md_df1.eq(md_df2)}")
print(f"pandas.eq:\n{pd_df1.eq(pd_df2)}")
# both print the following:
# modin/pandas.eq:
#        a      b      c
# 0  False  False  False
# 1  False  False  False

print(f"modin.__eq__:\n{md_df1 == md_df2}") # the same result as for `.eq`
print(f"pandas.__eq__:\n{pd_df1 == pd_df2}") # ValueError: dataframes has to be identically labeled

Issue Description

It seems that pandas have different logic for public and private boolean binary methods, while modin goes through the same method in both cases (query_compiler.eq).

Expected Behavior

We should match pandas here

Error Logs

Replace this line with the error backtrace (if applicable).

Installed Versions

Replace this line with the output of pd.show_versions()