Its a standard key for the Russian hatches.
bit short of time atm, will add more.
Actually answered here:
How is the docking hatch on the Progress/ATV opened?
The hatch opening procedure is described on page 39, section 5.2 of the Russian Segment Operations 2A.2B Flight document (PDF) posted on spaceref.com:
5.2. PROGRESS-СУ HATCH OPENING
Hatch tool→РАБОЧЕЕ ПОЛОЖЕНИЕ (Working Position)
Fully insert hatch tool extension into socket ГЕРМЕТИЗАЦИЯ КРЫШКИ (hatch actuator)
Secure hatch tool extension on SM passive docking unit cone
Fully insert hatch tool into socket on hatch tool extension
Rotate hatch tool in the direction of arrow ОТКРЫТИЕ (Open) until it clicks (6-7 turns)
Stow hatch tool and hatch tool extension into СтА accessories kit 11Ф732.Г4000А1-30
Open hatch and secure it in open position
Report to MCC. MCC-H⇒MCC-M
From the same link in the answer above i got most of the answers below:
What is it's function,
allows you to unlock, open and then remove the hatch.
how does it work?
Lots of gears. Not found a technical description or pictures.
Is it basically the keys to Nauka's front door?
Also keys to Progress, Soyuz, FGB, SM and PMA-1 doors...
When did it get to the ISS?
It is part of a docking accessories kit that includes wipes, extension tool and hatch cover. One kit went up on 2 November 2000 with first crew, unsure if it returned on the decent module though.
It would have been required then to open the hatch to enter Zvezda, FGB and their side of PMA-1.
After all the hatch openings it was returned to a docking accessories kit which was stored inside Soyuz.
So, my understanding is that each Soyuz has one, as they refer to a 'nominal position' it is kept in.
The others were launched with each module:
An identical kit is in a storage position on the axial transfer section on Zvezda.
(https://blogs.nasa.gov/stationreport/2020/11/20/iss-daily-summary-report-11-20-2020/ says: "Checking Docking Assembly (СтА) accessories kit stowage in SM ПрК" which is Zvezda's rear transfer tunnel)
Another kit is in storage location on panel 230 in FGB.
And another kit was tethered.to PMA-1 hatch handle, and later got moved to Unity module.
Nauka MLM probably has one so they do not need to keep going into the other modules to get parts from those kit bags.