Under the caveat that you haven't specified your embedded system, most TCP stacks will create a new socket for each new connection, and the mapping from socket to connection is 1-1.
When a packet arrives to the network stack, it has to associate that packet to the right socket. Usually, this is accomplished by employing a map from the TCP 4-tuple to the socket, where the 4-tuple consists of [local-ip, local-port, remote-ip, remote-port].
A server makes its service available by listening on a fixed local port that is known to clients wanting to use the service. As you understand, this is usually port 80 for a web server, and the software interface for most TCP implementations dedicate a socket for the purpose of allowing the API to perform operations on the network parameters for this service. However, the socket is not fully connected (the last two parts of the 4-tuple are set to a special "not specified" value, usually all bits 0). When a new connection is accepted, a new socket is created where the 4-tuple consists of the local information of the listening socket and the remote information taken from the source address and port of the SYN packet that initiated the TCP connection.
The limit on the number of connections a server can support is based on how the operating system is configured (you say yours limits it to 37). Using the 4-tuple, a single service (that is a fixed local-ip and local-port) will have an absolute limit of (2ADDR_BITS - RESERVED_ADDRS) × (216 - RESERVED_PORTS). For IPv4, the number of bits is 32, while for IPv6, the number of bits is 128.
When creating a connection, the client will specify the destination address and port (which fills out the remote information for the 4-tuple), but usually leave the source information unspecified. The TCP stack will choose an appropriate source address based on routing, and select an available source port (which will become the local information to complete the 4-tuple). In theory, any source port that is not being used by the selected local interface to communicate to the same remote service can be used as the local port. Most stacks will dedicate a set of the higher numbered ports for this purpose (referred to as the ephemeral port range).