Theoretically, yes. Practice, not. Most kernels (incl. linux) doesn't allow you a second bind()
to an already allocated port. It weren't a really big patch to make this allowed.
Conceptionally, we should differentiate between socket and port. Sockets are bidirectional communication endpoints, i.e. "things" where we can send and receive bytes. It is a conceptional thing, there is no such field in a packet header named "socket".
Port is an identifier which is capable to identify a socket. In case of the TCP, a port is a 16 bit integer, but there are other protocols as well (for example, on unix sockets, a "port" is essentially a string).
The main problem is the following: if an incoming packet arrives, the kernel can identify its socket by its destination port number. It is a most common way, but it is not the only possibility:
Because you are working on an application server, it will be able to do that.