The explanation has to be read as a whole to understand the context. It's about the body-and-soul duality.
Aging-and-death is a phenomena that visibly occurs to a body.
The two phrases are really the same because they imply that there is a body to which aging-and-death visibly occurs to, and that it is experienced by a soul.
If there is a separate body and soul, then this leads to eternalism and annihilationism, as explained in this answer.
This is an invalid question to ask, because it leads to eternalism and annihilationism, which renders it impossible to pursue the path to liberation ("living the holy life") because these are wrong views.
Instead, the Buddha taught the middle, which is dependent origination. This is the right view, conducive to pursuing the path to liberation.
When he had said this, a certain bhikkhu said to the Blessed One:
“Venerable sir, what now is aging-and-death, and for whom is there
this aging-and-death?”
“Not a valid question,” the Blessed One replied. “Bhikkhu, whether one
says, ‘What now is aging-and-death, and for whom is there this
aging-and-death?’ or whether one says, ‘Aging-and-death is one thing,
the one for whom there is this aging-and-death is another’— both these
assertions are identical in meaning; they differ only in the phrasing.
If there is the view, ‘The soul and the body are the same,’ there is
no living of the holy life; and if there is the view, ‘The soul is one
thing, the body is another,’ there is no living of the holy life.
Without veering towards either of these extremes, the Tathagata
teaches the Dhamma by the middle: ‘With birth as condition,
aging-and-death. ’”
SN 12.35