No. It is not a lie. It is the choice of words and how Yudhisthira projected him here, made the difference here.
First let's get to the analysis about Yudhisthira:
- He is said as lotus eyed
- He was borm on Jayeshta, second month of the year
- He was an expert in religion, science and administration
- His flag(yes Yudhisthira does have a flag on his chariot) is moon at the center and other planets around it
- One of the best master with Chariot and spear
- Owned a personal chariot with 1000 beautifully wrought tiger skins
Now look at the name he has given: Kanka
These are the characteristics of the name Kanka:
- Direct meaning - Scent of Lotus
- Astrologically, Kanka derives to number 2, with Moon as the ruling planet
Now let's compare notes. Yudhisthra has given him a pen name, Kanka because of his physical attributes(his eyes are like Lotus; but has to be hidden for an year. Only attribute of Lotus flower that is hidden from eyes is it's scent and it's long root. Yudhisthra took the scent for his name, which would still be hidden even if we search up to the roots) and his flag(He will rule under his flag and thus chosen Kanka as it is astrologically under the rule of moon. A name that can very well connect to his true identity, that is based on his physical attributes. A very well, strategically thought name.
Now one down, 2 more to go. Second part is his family name given Vaiyaghra.
He never informed the king that "My name is Kanka of Vaiyaghra family". Instead he informed the king "I am kanka, O king, I belong to the Vaiyaghra family." Now the meaning of Vaiyaghra? "car covered with a tiger's skin". So the indirect translation will be "kanka who belonged to tiger's skin covered car". That is something unique, at least in Mahabharatha, and is owned only by Yudhisthira. This you can read when Yudhisthira bets his car against Sakuni while dicing. From here:
Yudhishthira said,--'This my sacred and victorious and royal car which
gladdeneth the heart and hath carried us hither, which is equal unto a
thousand cars, which is of symmetrical proportions and covered with
tiger-skin, and furnished with excellent wheels and flag-staffs which
is handsome, and decked with strings of little bells,
Again, truth that was just hidden enough.
So somehow, two down. Only one to go. That is, he calling himself a Brahmana when he was a Kshatriya. The explanation for this one is a little stretch. One, is that Yudhisthira was not some one who upheld caste based on birth. He was arguing for Hidimbi to marry Bhima. An asura. And Pandavas were already living as Brahmanas for 12 full years. So Yudhisthira, until the return to rule, considering themself as Brahmana should be fine.
I do not have a link but in Nala-Damayandhi and Satyavan-Savithri stories, it was mentioned like if you live seven and a half years as friend or foe, or of a different caste, you become one of them. If this is from Vedas, Yudhisthira indeed knows them and might as well consider himself Brahmana at the exile.
As requested, I've added below how the other brothers and Draupadi named themselves to King Virata.
Bhima as Ballava
Ballava, directly means "A cow herd". Now get to Bhima's attributes: Super strong, one of the best in Mace/Wrestling and even a bigger eater. His guru, Balarama is the only one better than him at that point in terms of Strength and Power fighting. However, Bhima has no equal in terms of eating. It was highlighted in many places, even in Vana parva after Draupadi has Akshaya Patra. So he took his eating skill for disguise. Bhoja is the colloquial term that is used for eating and also synonymous with "Cow herd". To serve full to his appetite, as well as to prevent question about his caste(no one would question his caste, if he has a caste name as his name), he chose Ballava. The best is also to note that he never introduced his caste to King and due to the name, King also never questioned his caste. From here
Then the high-souled son of Pandu, approaching Virata, addressed him
in words that were not unsuited to his object, saying, 'O foremost of
kings, I am a cook, Vallava by name. I am skilled in dressing dishes.
Do thou employ me in the kitchen!'
Draupadi as Sairindhri
Again, a technique that is similar to Bhima yet went a little long further. Add a caste name as your name to avoid people asking your caste. And introduce your work as your name as well. She claims that I am a maid-servant of the Sairindhri class rather than saying "I belong to Sairindhri class/caste". We have to consider the meaning of "I am an independent artisan(hair dresser) who work as a maid-servant in other's house. She indeed told truth that she has five husbands(not a common thing even when polygamy is common) and also said "Draupadi would call her as malini", to give a name to the Queen Sudeshna.
If she is just lying, she indeed need not give true details(especially one where she says that she has 5 husbands)
Sahadeva as Vaisya
Now, this took me some time to understand why it is not a lie. Because Sahadeva introduce himself as
Sahadeva answered in accents deep as the roar of the cloud, 'I am a
Vaisya, Arishtanemi by name. I was employed as a cowherd in the
service of those bulls of the Kuru race, the sons of Pandu.
Instead of seeing Vaisya as a caste, we should see it as a continuous sentence with the meaning "some one trying to settle down". Like "I am a settler, Arishtanemi by name". Here Arishtanemi shows his role with Pandavas, the rim that holds the Pandavas with his knowledge. He also gives out his true talents and thus the point that he is lying is moot.
Arjuna as Vrihannala
No need to explain his name as he explains it with his introduction itself:
Know me, O king of men, to be Vrihannala, a son or daughter without
father or mother.'
This is a curse from Urvashi, it is right to call than Vrihannala(Not Arjuna, who has both father and mother) has only "mother"(cause of the curse) and thus took name as such.
Nakula as Granthika
Both the Ashvineyas(Nakul and Sahadeva) are extremely good at Astrology, fencing and ayurveda. And Nakul is more known for being extrememly handsome. Granthika, has 2 meanings, one of astrologist and other of "long pepper root". So Nakula just announced his mastery and chose the most beautiful Ayurvedic medicinal to match what is he famous for.
Basically, they all nearly announced their mastery or what they are known for, with their names. Best, we can call them as pen names instead of lies.