Is it necessary for one who entered the mosque late and prayed Isha thereafter to make up the six missed rak'ahs of Tarawih prayer?

It is not correct to pray Taraweeh before the 'Isha prayer; rather, it should be after the regular Sunnah of 'Isha. You could have joined the congregation with the intention of praying 'Isha, and when the Imam concludes after two rak'ahs, you would then stand to complete the remaining two rak'ahs. It is permissible for a Muslim to pray with the Taraweeh congregation with the intention of 'Isha, according to the soundest opinion of scholars. This is supported by the action of Mu'adh ibn Jabal - may Allah be pleased with him - who used to pray 'Isha with the Prophet, peace and blessings be upon him, and then lead his people in prayer, and the Prophet, peace and blessings be upon him, did not disapprove of that. The Prophet's prayer during the Prayer of Fear also supports this. As for making up any missed Taraweeh prayers, you have the option, because they are supererogatory prayers and not obligatory.