Having a SIN (fake or otherwise) does not seem to be a prerequisite for accessing the matrix or owning wireless enabled devices. But having a matrix persona (accessing the matrix) seem to be a prerequisite in order to take matrix actions, send instructions to wireless enabled devices you own and to broadcast your SIN (fake or otherwise).
If a corp figures out which persona hacked its host, it will still be able to pair you to that known persona tomorrow if given the opporunity. Your persona "individual identifiers" will remain identical a week from now.
It sure seem as if your devices somehow know who its legit owner is, even if the owner change icon or reboot their commlink a few times.
But it also seem as if hackers can only be tracked for as long as they remain online. While they are online they build up overwatch score. GOD might converge on them. People that was attacked with a data spike can take a matrix perception test to "spot the icon that just now attacked me with a data spike", even if the hacker is running silent.
Once the hacker reboot it seem as if there is no longer any evidence that it was they who hacked the host a few minutes ago. If it were then their OS would not reset and GOD would probably also automatically converge as they reconnect to the matrix. People that were hacked last week would be allowed to take a matrix perception test to "spot the same icon that hacked me last week". This does not seem to be the case.
why do fake SINs exist?
Corporations use SIN verification units to verify that you are a legit citizen.
...which a SINless person can defeat by using a fake SIN.
Fake SINs exist for SINless people that illegally wish to appear as if they are legit citizens.
To find out if you are you and if you belong a corporation typically use other security devices such as keypads, proximity RFID readers, ID badges, fingerprint scanners, palm print scanners, retinal print scanners, blood vessel pattern scanners, voice- and/or facial recognition, breath scanners, cellular scanners, DNA scanners...
...which you in return can defeat by hacking, rewiring circuits, using a sequencer, copy a legit owners card with a key card copier, forging ID badges, creating synthetic print glove-like membranes, using retinal duplication augmentations, voice recordings, voice modulators, prosthetic makeup and biosculpting or biometric samples from a legit user...
Stealing, copy, temporary borrowing or impersonating a specific individual's SIN does not seem to be supported in Shadowrun. And it doesn't seem to be needed either. If you wish to impersonate a legit citizen you just get a fake SIN. Same as stealing, copy, temporary borrowing or impersonating a specific individual's matrix persona does not seem to be supported in Shadowrun. And this does also not seem to be needed. If you wish to instruct the autopilot as if the instruction came from its legit owner then you need a mark on the owner and take the Spoof Command action from your own Persona.