Should IP-addresses paired with different subnet masks be seen as distinct?
No. A subnet mask is not a qualifier or namespace for IP addresses. Netmasks have no part in associating IP addresses with machines. Rather, they are associated with the physical and logical topology of the (IPv4) network, and they are needed for hosts to participate in the network appropriately. Using a different netmask does not change the meaning of IP addresses, so netmasks do not provide a mechanism for expanding the address space.
Does this mean that 192.168.0.1/24 and 192.168.0.1/16 are two different IP-addresses in practice?
No. In fact, those are not IP addresses at all -- they are (address, netmask) pairs, both with the same address part. On any given network, they refer to the same machine (if they refer to any machine at all). The IP address involved is in one of the non-routable ranges, however. These are usually used for internal networks, typically behind a router that performs network address translation (NAT) so that multiple machines can access the network without having globally-unique addresses. That has nothing to do with netmasks, however.
How many IP-addresses are there?
There are exactly 232 (a bit less than 5 billion) distinct IPv4 addresses. Not all of them are usable as host addresses. Use of private networks with NAT-ed access to the Internet expands the total number of machines that can be connected, but that does not change the number of distinct addresses, and it anyway is not related to netmasks.