UNMARKED SPOILERS AHEAD
When playing through Assassin's Creed: Syndicate, both Jacob and Evie Frye have missions in which they take out a special target (usually one of Starrick's more prominent henchmen). After they do them in, a certain scene follows where the background goes white (similar to the loading screen), the killer talks with their target as they lie dying, and they wipe blood off of the victim with a white handkerchief.
Example: when Evie kills Lucy Thorne (ends around 3:29).
However, in the scene where
he doesn't wipe the blood with the handkerchief. Why? And what is the greater symbolism and purpose of these "death scenes"?
Side note: if this is also present in other Assassin's Creed games or was explained then, it is outside of my knowledge because Syndicate is the first Assassin's Creed game I have played.
Best Answer
"What is the greater symbolism and purpose of these death scenes?"
Those death scenes are - in my opinion, as there seems to be no official explanation (as repeatedly lamented or critiqued in threads concerning this topic) - a condensation of the information collected (intentionally as well as intuitively) by the assassin, presented in a compressed and abstracted form in a virtual "white room".
This could have been done for the sake of gameplay (either ours or Desmond's, or both), as letting the player collect all this information would probably have been a tedious and time-consuming task, both for us and the developers.
The virtual room is either a purely symbolic device for us as players, or, in-universe, a kind of 'hub' where the information is relayed to Desmond.
Another way these scenes can be explained, and one that's not inconsistent with the last, is that the Assassin's supernatural Eagle Vision is able to create a telepathic link with the victim. After all, the assassin's do have supernatural abilities, and this might be one of them.
These death scenes are generally not (to be) seen as confessions, which is, for one, fairly obvious, considering the directness of the lethality of the signature hidden blade with which many of the victims are killed.
Besides: why would templars start spilling their guts as their guts are being spilled? After all, many of these victims were not repentant, stressing the earnestness of their ways.
As mentioned in the comments, wiping the blood of the victim with a feather was a way for Altaïr, and Bayek before him, to proof the killing to the Brotherhood.
These mannerisms of the Order of Assassins have disappeared or changed throughout the centuries: the Frye twins use their handkerchiefs - most other assassins don't bother, or take trophies from their victims.
On page 6 of Altaïr's Codex we can read the following:
And, after all (p. 4):
Ultimately, it doesn't matter how their ideologies are maintained.
"Why doesn't Jacob wipe the blood with the handkerchief?"
As for Jacob's abstaining when it comes to taking blood from Malcolm Millner, it is, like Wouter stated in the comments, unnecessary to prove this victim's death, as Jacob was misguided by Pearl Attaway who had ordered Millner's death, but was, unknown to Jacob at that time, involved with the Templars (source).