The documentation you have read was probably for Sitecore Commerce (XC) which is built using .net Core. You cannot use Serilog with Sitecore XP as XP is using log4net to do the logging.
However for your use case with New Relic, you can configure custom appender for log4net that will log Sitecore events to New Relic.
You need to create your custom log4net appender that will be using NewRelic.LogEnrichers.Log4Net npm package.
Then you need to configure appender similarly to other appenders that are present in Sitecore (check showconfig.aspx).
Similarly to this:
<log4net>
<appender name="NewRelicLoggingAppender" type="AssemblyName.Project.NewRelicAppender, AssemblyName.Project">
<file value="$(dataFolder)/logs/log.{date}.txt" />
<appendToFile value="true" />
<layout type="log4net.Layout.PatternLayout">
<conversionPattern value="%4t %d{ABSOLUTE} %-5p %m%n" />
</layout>
<encoding value="utf-8" />
</appender>
<logger name="AssemblyName.Project.NewRelicLogging" additivity="false">
<level value="INFO" />
<appender-ref ref="NewRelicLoggingAppender" />
</logger>
</log4net>