My simple goal is to have two instances. one is for database, second for just a site. Undoubtedly those 2 instances must have minimum network latency to communicate.
I know that there are so called placement groups but the problem is, they require expensive instance types to use.
Another option (to my mind) would be to use VPC. However, I can't use them. For some reasons in Management Console I can't create any VPC (look at the screenshot).
So my question is.
If I just create instances in the same region, would it work? I.e would they have a LAN like speed?
As long as the instances are in the same availability zone, latency will be very low. For normal uses, it'll be just fine.
You need to create one first, via the VPC console. You'll want to read the user guide closely, there's a lot of stuff involved in setting up a VPC.
That said, VPCs instances aren't located close to each other any more than regular EC2 ones. VPC affects networking setup, primarily.
That screenshot is not where you would go to create a VPC. You need to go to the VPC section of AWS, not the EC2 section, and create a VPC. Then create an Internet gateway. Then create a routing table and set it to route via the Internet gateway. Then create some subnets. Then, and only then, will you be able to create an instance in the VPC.