How should I customize UIPEthernet constants?
Creating a copy of the include file in sketch folder will not work. At library compilation the source files of sketch are not included. The &quot;&quot; for include is &quot;search first in the same folder as the including file&quot;.
The project in the description of use of uip-conf.h is UIPEthernet. It includes the UIP library by Adam Dunkels and configures it with this file.

A project with UIPEthernet library can configure the library with options in uipopt.h. Most of the options in uipopt.h are defined as
#ifdef UIP_CONF_UDP_CONNS
#define UIP_UDP_CONNS UIP_CONF_UDP_CONNS
#else /* UIP_CONF_UDP_CONNS */
#define UIP_UDP_CONNS    10
#endif /* UIP_CONF_UDP_CONNS */

this allows to define the value on compiler command line as -D. Most IDEs enable to set project defines in project settings. Arduino IDE does not enable this yet, because it doesn't have a project configuration file.
With Arduino IDE you can add defines for platform or for board definitions with build.extra_flags in platform.local.txt or in boards.local.txt. Example for boards.local.txt for Arduino Mega:
mega.build.extra_flags=-DUIP_CONF_UDP_CONNS=20 -DUIP_CONF_MAX_LISTENPORTS=30
platform.local.txt and boards.local.txt must be in boards package folder nect to platform.txt and boards.txt.