In brief, a given TCP connection is specified by four things:
- Host A IP address
- Host A port number
- Host B IP address
- Host B port number
This is what uniquely defines a given connection.
When a host opens a TCP connection to another (perhaps a web fetch), it chooses an unused port, typically a "random high port number", and makes the connection to the well-known port number of the server, perhaps 80. If it opens another connection to the same server, it will have a different local port.
Captured in real life during a web refresh of a page, my computer had many connections of which these are two:
As a given host has a maximum of 2^16 ports (in IPv4), it can have at most 65,536 simultaneous connections to a given web server on the normal web port (minus some ports which have special meanings). That's not a problem for an individual's computer, but might be for a large web proxy. (And a given operating system might well have other much smaller limits: the limit I'm speaking of is inherent in the protocol limit.)
Hope that helps clarify somewhat.