2 separate IP addresses

Two separate ip addresses for your computers or are you asking if you can have two separate ip addresses for two local networks, or two separate public ip addresses?

If for the computers on the same network, they must have their own ip addresses. These are typically obtained from your lan router using DHCP.

If you mean twi weparate public ip addresses then you must talk with your isp and get two pulic ip addresses and likely two modems and two accounts.

If you mean two lan ip addresses then your router must be upgraded to something that can administer more than one lan.

You need a router. Connect each computer to the router using either ethernet or wifi. The router will give each computer its own ip address.

You need - and may already have - a router (you can't just connect two computers across the same circuit). Each will have its own IP number on the local network (often beginning 192...). However as far as the outside world is concerned both have the same IP number, the one allocased by your ISP. The router recognizes which of your computers is making the request, and sends the returning packets to the correct machine, but that difference is not externally visible.

So if your girlfriend is trying, for example, to log into a site which expects every login to have a unique IP number, and you have already logged into it, the site wouldn't accept her login. However it's fairly unusual for this sort of situation to arise - are you sure it's the requirement for a unique IP number which is causing the problem? The only way you can get different external IP numbers for the two of you is to have two separate accounts with your ISP, which might very well also require two separate physical lines.

As far as the sites are concerned both your computers have the IP number allocated to your router (often a combined modem/router) by your ISP. They cannot see the IP numbers which the router allocates to your Macs. You cannot do this unless you have a completely separate internet connection with its own separate external IP number (a mobile phone type of connection would satisfy this requirement - you could use a mobile internet 'dongle').

The site sees both your computers as having the same IP.

Your ISP allocates an IP number to your modem/router. This is the number that the outside world sees.

The router allocated local IP numbers to all devices connected to it. It recognizes which device has placed a request and routed the incoming requested data to it. The outside world cannot see these numbers (the same range of numbers is used by millions of people but always confined within their local network).

If a site requires the two of you to have separate IPs you cannot both join it.