If we view their relationship as Guru and Disciple, then we will get different idea. It is a well known fact that a teacher will develop affection towards some students only, mainly due to their grasping power of the subject.
Vidura was known to be embodiment of Dharma. Out of 5 Pandavas and 100 Kauravas, he had shown affection towards Yudhisthira only, because the latter also embodiment of Dharma.
We have to remember that it was Vidura, who cautioned only Yuddhisthira in Lakshagriha episode. Vidura was the Guru and Yudhisthira was his disciple.
That's why Vidura transmitted his SPIRITUAL energy to Yidhisthira in his last moments.
We should also remember that it was Yidhisthira alone went up to the pinnacle of Himayala during mahaprasthAna.
We will come across cases of Guru transmitting his energy to his disciple just before the death of the former.
The following is from the life history of Sri Ramakrishna Paramahamsa.
In last days before Sri Ramakrishna's departure, Narendra being alone
with the Master, Sri Ramakrishna looked at him and went into samadhi.
Narendra felt the penetration of a subtle force and lost all outer
consciousness. Regaining presently the normal mood, he found the
Sri Ramakrishna said to him: "Today I have given you my all and I am
now only a poor fakir, possessing nothing. By this power you will do
immense good in the world, and not until it is accomplished will you
return." Henceforth the Master lived in the disciple.