This is "Let's Be Frank" by Brian Aldiss. An online copy (with terrible formatting) is available.
Strangely perturbed, as if something inexpressible lay just below the surface of his thoughts, Sir Frank went over to the bed and rested his hands on the boy's shoulders. He seemed to stand on the brink of a precipice.
"Frank," he whispered—for the sleeping child had been given his own name—"Frank, why don't you wake up?"
In answer to the words, the youth's eyes opened. The usual wash of dizziness came and went like a flash; Sir Frank found himself looking up into his own eyes. [...]
"What sorcery is this?" Sir Frank muttered.
But it was no sorcery, or not in the sense Sir Frank meant. He had merely acquired an additional body for his ego. It was not that he could be in either as he pleased; he was in both at the same time. When the son came finally to consciousness, it was to his father's consciousness.