As I know, generally there are two types of IXPs
This is incorrect.
type 1: use exchange routers, which works in layer 3
This is not an IXP. This is a router. That router would be owned by someone, who would have some sort of policy in the router, which would make it an Internet service provider, not an Internet exchange point.
type 2: use switches and Ethernet topology, which works in layer 2.
This is an IXP. Routers belonging to Internet service providers, communicating with each other across a switch fabric, which is an Internet exchange point.
1. For type 1, the exchange routers may use several IP prefixes for routing,
how often does the IP prefixes have their own AS?
Since this is not an IXP, I think the question is irrelevant to your research.
If an ISP wants to participate in BGP routing, and originate an IP prefix, that ISP must have an AS.
2. For type 2, all peers connected to the IXP must work in the same subnet
required by Ethernet rules.
Generally, yes, though some IXPs are not that prescriptive, and would allow a subset of the ISPs to peer on a different subnet if they wished.
Is possible that the subnet IP prefixes belong to some private IP address space, such as 192.168.x.x?
It is possible, but it does not follow best-practices, because it breaks traceroute and other diagnostic tools.
How often does this happen?
Very very rarely.
Only two IXPs out of more than three hundred are using FRC1918 space at this point: Maputo and Santiago de Compostela.
This used to be a more common mistake, but as communications with the operators of new IXPs has improved over time, it's become very rare.
If the subnet only contains public IP addresses, how are the addresses announced?
They are generally not announced. Occasionally they're announced by one or more participating ISPs at the IXP. Sometimes that's purposeful, other times it's accidental. Some IXPs have rules prohibiting the announcement of the exchange subnet, others actively seek out sources of transit for the exchange subnet.
Packet Clearing House