I think I've found one possible answer, so I will answer my own question, but I will not accept it.
The sutta quoted in the question states that the eyes, ears, nose, tongue, body and mind are old kamma, but it doesn't say WHOSE old kamma it is.
The sutta below answers that question.
So, the eyes, ears, nose, tongue, body and mind definitely arise due to old kamma, as stated in ChrisW's answer. After all, they cannot arise spontaneously - they definitely arise due to old kamma. But we cannot ask WHOSE old kamma it is. It is dependently originated - with ignorance as the condition.
This is in line with sabbe dhamma anatta (all phenomena is not self).
From SN 12.17:
“How is it, Master Gotama: is suffering created by oneself?”
“Not so, Kassapa,” the Blessed One said.
“Then, Master Gotama, is suffering created by another?”
“Not so, Kassapa,” the Blessed One said.
“How is it then, Master Gotama: is suffering created both by oneself
and by another?”
“Not so, Kassapa,” the Blessed One said.
“Then, Master Gotama, has suffering arisen fortuitously, being created
neither by oneself nor by another?”
“Not so, Kassapa,” the Blessed One said.
“How is it then, Master Gotama: is there no suffering?”
“It is not that there is no suffering, Kassapa; there is suffering.”
“Then is it that Master Gotama does not know and see suffering?”
“It is not that I do not know and see suffering, Kassapa. I know
suffering, I see suffering.”
“Whether you are asked: ‘How is it, Master Gotama: is suffering
created by oneself?’ or ‘Is it created by another?’ or ‘Is it created
by both?’ or ‘Is it created by neither?’ in each case you say: ‘Not
so, Kassapa.’ When you are asked: ‘How is it then, Master Gotama: is
there no suffering?’ you say: ‘It is not that there is no suffering,
Kassapa; there is suffering.’ When asked: ‘Then is it that Master
Gotama does not know and see suffering?’ you say: ‘It is not that I do
not know and see suffering, Kassapa. I know suffering, I see
suffering.’ Venerable sir, let the Blessed One explain suffering to
me. Let the Blessed One teach me about suffering.” “Kassapa, if one
thinks, ‘The one who acts is the same as the one who experiences the
result,’ then one asserts with reference to one existing from the
beginning: ‘Suffering is created by oneself.’ When one asserts thus,
this amounts to eternalism. But, Kassapa, if one thinks, ‘The one who
acts is one, the one who experiences the result is another,’ then one
asserts with reference to one stricken by feeling: ‘Suffering is
created by another.’ When one asserts thus, this amounts to
annihilationism. Without veering towards either of these extremes, the
Tathagata teaches the Dhamma by the middle: ‘With ignorance as
condition, volitional formations come to be; with volitional
formations as condition, consciousness…. Such is the origin of this
whole mass of suffering. But with the remainderless fading away and
cessation of ignorance comes cessation of volitional formations; with
the cessation of volitional formations, cessation of consciousness….
Such is the cessation of this whole mass of suffering.’”