log4j - write to file name as the logger name

You can't do it like that with the RollingFileAppender. The appender receives log events as configured by your <Loggers> block, if log4j sends it log events with different LoggerNames they will be appended to whatever file is open. A RollingFileAppender writes to one file at a time and rolls when the configured policy tells it to.

You could write to different files by setting up multiple Loggers that target different appenders. Like this:

<Configuration status="info">
    <Appenders>
        <Console name="Console" target="SYSTEM_OUT">
            <PatternLayout pattern="%d{dd.MM.yyyy HH:mm:ss.SSS} %-5level %logger - %msg%n"/>
        </Console>
        <RollingFile name="FileA"
                     filePattern="/tmp/A-%d{yyyy.MM.dd_HH.mm.ss}-%i.log">
            <PatternLayout pattern="%d{dd.MM.yyyy HH:mm:ss.SSS} %-5level %logger - %msg%n"/>
            <SizeBasedTriggeringPolicy size="1 MB"/>
        </RollingFile>
        <RollingFile name="FileB"
                     filePattern="/tmp/B-%d{yyyy.MM.dd_HH.mm.ss}-%i.log">
            <PatternLayout pattern="%d{dd.MM.yyyy HH:mm:ss.SSS} %-5level %logger - %msg%n"/>
            <SizeBasedTriggeringPolicy size="1 MB"/>
        </RollingFile>
    </Appenders>
    <Loggers>
        <Logger name="org.example.App" level="info" additivity="false">
            <AppenderRef ref="FileA"/>
        </Logger>
        <Logger name="org.example.App2" level="info" additivity="false">
            <AppenderRef ref="FileB"/>
        </Logger>
        <Root level="info">
            <AppenderRef ref="Console"/>
        </Root>
    </Loggers>
</Configuration>

To route each event to a different RollingFileAppender based on some pattern you can use a RoutingAppender.

The RoutingAppender evaluates LogEvents and then routes them to a subordinate Appender. The target Appender may be an appender previously configured and may be referenced by its name or the Appender can be dynamically created as needed. The RoutingAppender should be configured after any Appenders it references to allow it to shut down properly.

You can also configure a RoutingAppender with scripts: you can run a script when the appender starts and when a route is chosen for an log event.

Here is how you could route based on LoggerName:

<Configuration status="info">
    <Properties>
        <Property name="log-path" value="logs/"/>
        <Property name="file-name" value="server"/>
        <Property name="file-type" value=".log"/>
    </Properties>
    <Appenders>
        <Console name="Console" target="SYSTEM_OUT">
            <PatternLayout pattern="%d{dd.MM.yyyy HH:mm:ss.SSS} %-5level %logger - %msg%n"/>
        </Console>
        <Routing name="Routing">
            <Routes pattern="$${event:Logger}">
                <Route>
                    <RollingFile name="Rolling-${event:Logger}" fileName="${log-path}${file-name}-${event:Logger}${file-type}"
                                 filePattern="${file-name}-%d{yyyy.MM.dd_HH.mm.ss}-%i-${event:Logger}.log">
                        <PatternLayout pattern="%d{dd.MM.yyyy HH:mm:ss.SSS} %-5level %logger - %msg%n"/>
                        <SizeBasedTriggeringPolicy size="1 MB"/>
                    </RollingFile>
                </Route>
            </Routes>
        </Routing>
    </Appenders>
    <Loggers>
        <Root level="info">
            <AppenderRef ref="Routing"/>
            <AppenderRef ref="Console"/>
        </Root>
    </Loggers>
</Configuration>

Also, out of interest, if you really want to you can implement your own Appender that does whatever you want with the LogEvent.

Here's a very rough start. It's an Appender that creates a RollingFileAppender per LoggerName. It only works if you use a SizeBasedTriggeringPolicy. You should be able to drop a class like that into your project and use XML like:

<Configuration status="info">
    <Appenders>
        <Console name="Console" target="SYSTEM_OUT">
            <PatternLayout pattern="%d{dd.MM.yyyy HH:mm:ss.SSS} %-5level %logger - %msg%n"/>
        </Console>
        <FilePerLoggerNameAppender name="File"
                     filePattern="/tmp/log-%d{yyyy.MM.dd_HH.mm.ss}-%i-$LOGGER$.log">
            <PatternLayout pattern="%d{dd.MM.yyyy HH:mm:ss.SSS} %-5level %logger - %msg%n"/>
            <SizeBasedTriggeringPolicy size="1 MB"/>
        </FilePerLoggerNameAppender>
    </Appenders>
    <Loggers>
        <Root level="info">
            <AppenderRef ref="File"/>
            <AppenderRef ref="Console"/>
        </Root>
    </Loggers>
</Configuration>

and see log files created with $LOGGER$ replaced with your logger names.