A lost little boy is always a lost little boy at heart.
I love Wilf so much because while the Doctor also needed his companions, they needed him just as much. Except for Wilf. Wilfred had seen life, lived a long life, and although he wasn’t a centuries-old Timelord, was to all intents and purposes the older of the two. He was the one offering the exact support the Doctor needed when he, as an old man, was ordered the impossible: not to die. It was the sort of reassurance of the impossible that parents give to their kids when they realize all they need is even just a token shelter from reality.
I get the distinct impression that the Doctor’s father was at the least neglectful and dismissive, if not an emotionally abusive arseface. And the Doctor and Wilfred recognized each other. Wilfred saw a terrified, brave kid, and the Doctor saw the dad he never had.