It is just a folklore and is not present in any scriptures of Hinduism. Dhritarashtra didn't arrest Shakuni and his family.
The story you told is present across internet 1, 2. You can observe that they are not quoting any scriptures and are clearly saying that the story is not present in any Hindu scripture.
We can prove the falsity of the story with the help of the fact that Suvala, father of Shakuni, attended the Rajasuya sacrifice performed by Yudhishthira.
Vaisampayana said,--"the ever-victorious Nakula, the son of Pandu,
having reached Hastinapura, formally invited Bhishma and
Dhritarashtra. The elder of the Kuru race with the preceptor at their
head, invited with due ceremonies, came with joyous hearts to that
sacrifice, with Brahmanas walking before them. And, O hull of the
Bharata race, having heard of king Yudhishthira's sacrifice, hundreds
of other Kshatriyas acquainted with the nature of the sacrifice, with
joyous hearts came there from various countries, desiring to behold
king Yudhishthira the son of Pandu and his sacrificial mansion, and
brought with them many costly jewels of various kinds. And
Dhritarashtra and Bhishma and Vidura of high intelligence; and all
Kaurava brothers with Duryyodhana at their head; and Suvala the king
of Gandhara and Sakuni endued with great strength; and Achala, and
Vrishaka, and Karna that foremost of all charioteers; and Salya endued
with great might and the strong Valhika; and Somadatta, and Bhuri of
the Kuru race, and Bhurisravas and Sala; and Aswatthama, Kripa, Drona,
and Jayadratha, the ruler of Sindhu; and Yajnasena with his sons, and
Salya that lord of earth and that great car warrior king Bhagadatta of
Pragjyotisha accompanied by all Mlechcha tribes inhabiting the marshy
regions on the sea-shore; and many mountain kings, and king
Vrihadvala; and Vasudeva the king of the Paundrayas, and the kings of
Vanga and Kalinga; and Akastha and Kuntala and the kings of the
Malavas and the Andhrakas; and the Dravidas and the Singhalas and the
king of Kashmira, and king Kuntibhoja of great energy and king
Gauravahana, and all the other heroic kings of Valhika; and Virata
with his two sons, and Mavella endued with great might; and various
kings and princes ruling in various countries; and, O Bharata king
Sisupala endued with great energy and invincible in battle accompanied
by his son--all of them came to the sacrifice of the son of Pandu. And
Rama and Aniruddha and Kanaka and Sarana; and Gada, Pradyumna, Shamva,
and Charudeshna of great energy; and Ulmuka and Nishatha and the brave
Angavaha; and innumerable other Vrishnis--all mighty
car-warriors--came there.
[Section 33, Rajasuyika Parva, Sabha Parva, The Mahabharata]
According to the folklore stories mentioned over internet, Shakuni's father was dead and hence cannot attend the sacrifice with Shakuni. Therefore, the stories are not true according to the Mahabharata.