Could you please help me identify what the vendor and type of this connector? I have only managed to find it on aliexpress named as PHB 2.0mm Connector. But I suspect it to be a copy of a JST or other brand.
I need to know which crimp tool I can use for it.
It's made by many companies under the generic name of "PHB".
My Identiconn lists some:
https://connectorbook.com/identification.html?N=&n=lo_prof_w2b_conn&c=DL20006&p=U
They include DLL: 20006; HCTL: HC-PHB; Megastar: ZX-PHB2.0; XunPu: WAFER-PHB2.0; DLL: PHB; Hanbo Electronic: WAFER-PHB200; JKUN: A2002 A2004; XunDa: XD-PHB;
Also:
YeOnho: https://connectorbook.com/identification.html?N=&n=lo_prof_w2b_conn&c=SMx200-NNC&p=U
HDGC: https://connectorbook.com/identification.html?N=&n=lo_prof_w2b_conn&c=HDGC2006&p=U
Konnra: https://connectorbook.com/identification.html?N=&n=lo_prof_w2b_conn&c=KR2012&p=U
{Source: DLL}
LCSC sells them:
I need to know which crimp tool I can use for it.
These kinds of manufacturers do not specify tooling. They expect you to buy wire tails with ready-crimped contacts, or to use generic crimping tools.
From what I gather, it seems to be some 2mm pitch connector perhaps originating from JST PH series, but is not made by JST nor compatible with JST. "PHB 2.0" is probably an attempt to make it sound like a legit JST and not some cheap clone. Something called "YEONHO SMH200" also pops up, example:
https://www.klsconnector.com/2-00mm-pitch-yeonho-smh200-double-with-lock-wire-to-board-connector/
I have no idea if the crimping is compatible with JST PH or not. There is probably no reason why you would use these instead of buying proper JST connectors, especially if you also need to buy crimping tools.
For a small scale project, some careful work with a small pair of pliers will be fine - no need for a specific crimping tool if you don't have it.
If you want to spend money on tools, the crimping die you want is something like this: 
This crimping die ($18) grabs the two tabs on your connector and fold them in so their tips are pointed inwards/down, as opposite to just wrapping one on the top of the other.
Note there are two sets of tabs on each pin: the rear one is supposed to be crimped to the insulation and act as strain relief while the smaller ones are what bites/crimps the exposed wire.
The crimping tool looks like this ratcheting one. You can buy a bunch of different dies and swap them to fit your needs. As someone said earlier, for a one-off it does not make sense (tool plus one single die ~$50), but I have those for more than a decade and they still deliver crimps that look like what you see used by factory in cars. I think they beat the quintessential "pig-tail" you find at auto parts stores, which is then attached to the old wire with a splice of sorts.
