Your question is lacking quite a bit of detail before a specific answer can be crafted. A diagram of your current equipment setup would help.
Given your question is about how basic routing works and you mention both locations have WiFi, i am going to assume that your network consists of 2 wireless routers, both are performing NAT.
I have tried to minimize the technical terms, since i believe it would confuse you, however the price of this decision is ambiguity and lack of detail. Regardless i think this will serve you better.
Here is a simple diagram of what i assume your network looks like:
Is it possible for a PC/smartphone on the remote site to be assigned a private IP (192.168.2.x) and use it for both Internet and Intranet access? If yes, how?
Answer 1: Intranet access
yes it is possible.
Once the client device associates with the remote location AP/Router, the DHCP server in that device will issue an IP address to the client. It will also specify a default gateway for the client to use.
On the diagram i show the client device has been assigned IP: 192.168.2.2 and gateway: 192.168.2.1.
You need to know the function of the default gateway to understand how the client can access other networks. Here is how it works. The client can talk DIRECTLY to any other host that is on the LAN side of the remote location AP/Router. That is any host with an IP address in the range: 192.168.2.1-192.168.2.254.
Now the client (a host) with IP: 192.168.2.2 wants to access your file server (another host) with IP: 192.168.1.100.
Clearly 192.168.1.100 is not part of the "192.168.2.1-192.168.2.254" range. As a result the client cannot talk directly to the file server. Anytime a host cannot talk directly to another it uses the default gateway address (oversimplification, but routing tables is outside the scope of this question), in this case 192.168.2.1.
Now the request to get to 192.168.1.100 arrives at the remote location AP/Router, because its LAN side IP acts as the default gateway for the client. The router looks at who the client wants to talk to, then looks at the IP ranges the AP/router can talk directly to. The router has 2 ranges, one on the WAN side and one on the LAN side. The router can talk directly to the following ranges: "192.168.1.1-192.168.1.254" and "192.168.2.1-192.168.2.254".
There is a match! 192.168.1.100 is inside the "192.168.1.1-192.168.1.254" range, the router now relays the request to the file server.
Answer 1: Internet access
Again yes it is possible.
Same example as above, however instead of requesting data from the file server (192.168.1.100) the client wants access to Google. Google has IP: 22.214.171.124.
The client device compares and determines that 126.96.36.199 is outside any range it can talk to directly, send the request to its default gateway (remote AP/Router).
The remote AP/Router does the same comparison. Before it found that it could talk directly to the file server, however in this case it cannot talk directly to Google. So it does the same as the client, it forwards the request to its default gateway, in this case the default gateway (192.168.1.1) is the LAN side of the "local AP/Router".
Now the "local AP/Router" receives the request and does the same comparison. It also realizes that it cannot talk directly to 188.8.131.52 and send the request on to the default gateway it received from the ISP.
And this way the request from the client keeps bouncing from router to router until it reaches a router that can talk directly to Google.
Because i assume your setup is quite simple it is important to note that if the client and file server switched places the client would not be able to reach the file server. I.e. client associated with "local AP/Router" and file server connected on the LAN of the "remote AP/Router". This is because "local AP/Router" has no knowledge of the 192.168.2.1-192.168.2.254 range and would as a result relay the request to the ISP ("away" from the file server), who will drop the request.
Considering that the remote site and main office has different address range, is it possible for the Intranet to even exist in the first place?
Like others have commented, an intranet is simply a network where the resources (like a file server) can only be accessed from within the network. That is opposed to the Internet where the resources are available to the public.
The amount of sub nets is irrelevant.