Multiport Server NIC ports?
Solution 1:
These devices are typically seen in the OS as four separate network devices and they can be used just like the four separate NICs you had before.
Solution 2:
As others have already mentioned, yes, a 4-port network card will be recognized and usable just like 4 separate network interfaces.
However, in particular with 4-port cards, you should pay attention to the bus bandwidth the card is capable of. For example, the Intel EXPI9404VT in your second link has a PCIe 1.1 x4 bus interface, meaning that it has a total bandwidth of about 1 gigabyte per second in each direction. That's plenty for four 1-gigabit network interfaces.
Sometimes there are cheaper multi-port network cards whose bus bandwidth is less than the total bandwidth of their network ports. These can be useful if you need only connectivity to several networks, not sustained full performance. With today's PCIe bus, you're not too likely to come across this until you work with 10Gig network ports, but it's a thing to keep in mind if you need a lot of network ports in an older server.
Solution 3:
yes it will be recognized as 4 different ports, with 4 different mac addresses, and you should see 4 interfaces where you can assign 4 IPs