Nibbana is that which the mind experiences when it is free of defilements, free of the three poisons of greed, aversion and delusion. It is not a state of mind or consciousness. It is certainly not a self or a witness. Nibbana is not conditioned as in it is not caused or created by anything, or composed of anything, or influenced by anything. On the other hand, states of mind or consciousness is conditioned and changing as time goes on.
AN 9.34 states that nothing is felt or sensed about Nibbana and that's precisely why it is blissful. The absence of suffering (dukkha) is blissful. It's not blissful because it is felt by the senses to be pleasurable - that would make it conditioned. This answer explains more on Nibbana and bliss.
According to Iti 44, Nibbana-element without residue left describes the five aggregates in parinibbana, based on Ven. Thanissaro's footnote. This means that all five aggregates, including consciousness, stop operating completely. In this case, all mental fabrications (sankhara) have been stilled.
Nibbana-element with residue left describe the the five aggregates of a living Arahant i.e. the five aggregates (form, feeling, perception, consciousness, mental formations) are functioning without craving, clinging and defilements. The flames of passion, aversion and delusion are no more burning for the living Arahant. But the five aggregates function without clinging, like glowing embers, for the living Arahant. In this case, only the clinging mental fabrications (sankhara) have been stilled. This question discusses clinging aggregates vs. non-clinging aggregates. In parinibbana, the glowing embers are put out (Nibbana-element without residue left).
So, you can see that that when the Arahant is living, the clinging sankhara (mental fabrications) are stilled. In parinibbana, all sankhara (mental fabrications) are stilled.
What's interesting is that all mental fabrications (one meaning of sankhara) are conditioned (another meaning of sankhara), but not all that is conditioned are mental fabrications. For e.g. water is conditioned, but it's not a mental fabrication (unless you refer to the mental concept of water).
So, in my opinion, the unconditioned (asaṅkhataṃ) in Ud 8.3 may refer to all that is unconditioned. But "the stilling of the conditioned is bliss" (saṅkhārūpasamo sukho”ti) from Iti 43, specifically refers to clinging mental fabrications and not the stilling of everything that is conditioned.
The born, come-to-be, produced,
The made, the conditioned, the transient,
Conjoined with decay and death,
A nest of disease, perishable,
Sprung from nutriment and craving's cord —
That is not fit to take delight in.
The escape from that, the peaceful,
Beyond reasoning, everlasting,
The not-born, the unproduced,
The sorrowless state that is void of stain,
The cessation of states linked to suffering,
The stilling of the conditioned — bliss.
Iti 43 (translated by Ireland)
Interestingly, Ven. Thanissaro used the word "fabrications" instead of "conditioned" in his translation. I would go further to call it "clinging fabrications".
The stilling of the conditioned (i.e. clinging mental fabrications) is bliss.
The born, become, produced,
made, fabricated, impermanent,
composed of aging & death,
a nest of illnesses, perishing,
come from nourishment
and the guide [that is craving] —
is unfit for delight.
The escape from that is
the sorrowless, stainless state,
the cessation of stressful qualities,
the stilling of fabrications, bliss.
Iti 43 (translated by Thanissaro)