Vidura is not the father of Yudhishthira. The circumstances of Yudhishtira's birth are described in this chapter of the Adi Parva of the Mahabharata:
O Janamejaya, when Gandhari's conception had been a full year old, it was then that Kunti summoned the eternal god of justice to obtain offspring from him. And she offered without loss of time, sacrifices unto the god and began to duly repeat the formula that Durvasa had imparted to her some time before. Then the god, overpowered by her incantations, arrived at the spot where Kunti was seated in his car resplendent as the Sun. Smiling, he asked, 'O Kunti, what am I to give thee?' And Kunti too smiling in her turn, replied, 'Thou must even give me offspring.' Then the handsome Kunti was united (in intercourse) with the god of justice in his spiritual form and obtained from him a son devoted to the good of all creatures. And she brought his excellent child, who lived to acquire a great fame, at the eighth Muhurta called Abhijit, of the hour of noon of that very auspicious day of the seventh month (Kartika), viz., the fifth of the lighted fortnight, when the star Jyeshtha in conjunction with the moon was ascendant. And as soon as the child was born, an incorporeal voice (from the skies) said, 'This child shall be the best of men, the foremost of those that are virtuous. Endued with great prowess and truthful in speech, he shall certainly be the ruler of the earth. And this first child of Pandu shall be known by the name of Yudhishthira. Possessed of prowess and honesty of disposition, he shall be a famous king, known throughout the three worlds.
Now as to the Upanishad reference, Karve is referring to the ritual described in Adhyaya 2 of the Kaushitaki Upanishad:
Next follows the father's tradition to the son, and thus they explain it. The father, when going to depart, calls his son, after having strewn the house with fresh grass, and having laid the sacrificial fire, and having placed near it a pot of water with a jug (full of rice), himself covered with a new cloth, and dressed in white. He places himself above his son, touching his organs with his own organs, or he may deliver the tradition to him while he sits before him. Then he delivers it to him. The father says: 'Let me place my speech in thee.' The son says: 'I take thy speech in me.' The father says: 'Let me place my scent (prâna) in thee.' The son says: 'I take thy scent in me.' The father says: 'Let me place my eye in thee.' The son says: 'I take thy eye in me.' The father says: 'Let me place my ear in thee.' The son says: 'I take thy ear in me.' The father says: 'Let me place my tastes of food in thee.' The son says: 'I take thy tastes of food in me.' The father says: 'Let me place my actions in thee! The son says: 'I take thy actions in me.' The father says: 'Let me place my pleasure and pain in thee.' The son says: 'I take thy pleasure and pain in me.' The father says: 'Let me place happiness, joy, and offspring in thee.' The son says: 'I take thy happiness, joy, and offspring in me.' The father says: 'Let me place my walking in thee.' The son says: 'I take thy walking in me 1.' The father says: 'Let me place my mind in thee.' The son says: 'I take thy mind in me.' The father says: 'Let me place my knowledge (pragñâ) in thee.' The son says: 'I take thy knowledge in me.' But if the father is very ill, he may say shortly: 'Let me place my spirits (prânas) in thee,' and the son: 'I take thy spirits in me.' Then the son walks round his father keeping his right side towards him, and goes away. The father calls after him: 'May fame, glory of countenance, and honour always follow thee.' Then the other looks back over his left shoulder, covering himself with his hand or the hem of his garment, saying: 'Obtain the heavenly worlds (svarga) and all desires.' If the father recovers, let him be under the authority of his son, or let him wander about (as an ascetic). But if he departs, then let them despatch him, as he ought to be despatched, yea, as he ought to be despatched.
A similar ritual is described in Adhyaya 1 Brahmana 5 of the Brihadaranyaka Upanishad
Next follows the handing over. When a man thinks he is going to depart, he says to his son: 'Thou art Brahman (the Veda, so far as acquired by the father); thou art the sacrifice (so far as performed by the father); thou art the world.' The son answers: 'I am Brahman, I am the sacrifice, I am the world.' Whatever has been learnt (by the father) that, taken as one, is Brahman. Whatever sacrifices there are, they, taken as one, are the sacrifice. Whatever worlds there are, they, taken as one, are the world. Verily here ends this (what has to be done by a father, viz. study, sacrifice, &c.) 'He (the son), being all this, preserved me from this world,' thus he thinks. Therefore they call a son who is instructed (to do all this), a world-son (lokya), and therefore they instruct him. When a father who knows this, departs this world, then he enters into his son together with his own spirits (with speech, mind, and breath). If there is anything done amiss by the father, of all that the son delivers him, and therefore he is called Putra, son. By help of his son the father stands firm in this world. Then these divine immortal spirits (speech, mind, and breath) enter into him.
Now granting the premises, why would Vidura interact with Yudhishthira in a manner akin to a father and son, when Vidura was not the father of Yudhisthira? Because Vidura was an incarnation of Yudhisthira's father Yama, as described in this chapter of the Adi Parva of the Mahabharata:
And, O king, thou shouldst know that he who was known on earth as Vidura, who was the first of all virtuous men, who was the god of Justice himself[.]... And, O monarch, learn that king Yudhishthira was a portion of Dharma; that Bhimasena was of the deity of wind; that Arjuna was of Indra, the chief of the celestials; and that Nakula and Sahadeva, the handsomest beings among all creatures, and unrivalled for beauty on earth, were similarly portions of the twin Aswins.
That is the point Vyasa is making in the quote misinterpreted by Karve from this chapter of the Ashramavasika Parva of the Mahabharata:
The eternal deity of Righteousness was stupefied by the Rishi Mandavya with an expenditure of his penances earned for a long time with great care. At the command of the Grandsire, and through my own energy, Vidura of great intelligence was procreated by me upon a soil owned by Vichitraviryya. A deity of deities, and eternal, he was, O king, thy brother. The learned know him to be Dharma in consequence of his practices of Dharana and Dhyana. He grows with (the growth of) truth, self-restraint, tranquillity of heart, compassion, and gifts. He is always engaged in penances, and is eternal. From that deity of Righteousness, through Yoga-puissance, the Kuru king Yudhishthira also took his birth. Yudhishthira, therefore, O king, is Dharma of great wisdom and immeasurable intelligence. Dharma exists both here and hereafter, and is like fire or wind or water or earth or space. He is, O king of kings, capable of going everywhere and exists, pervading the whole universe. He is capable of being beheld by only those that are the foremost of the deities and those that are cleansed of every sin and crowned with ascetic success. He that is Dharma is Vidura; and he that is Vidura is the (eldest) son of Pandu. That son of Pandu. O king, is capable of being perceived by thee. He stays before thee as thy servitor. Endued with great Yoga-puissance, thy high-souled brother, that foremost of intelligent men, seeing the high-souled Yudhishthira, the son of Kunti, has entered into his person.