make log_sending=true correlate to the correct service

## Is your feature request related to a problem?
This issue is based on my post on Discuss: https://discuss.elastic.co/t/room-for-improvement-for-log-sending-true/330602/1

We have a Tomcat container running with multiple applications. Each application is a Spring webapplication deployed as a War-file. we use the apm agents service-name-auto-detection to get the correct `service.name` for each app. this looks like this:
- tomcat-application
- App 1
- App 2
- ...

When activating `log_sending`, the logs are transferred without errors to the Elastic Stack, but the Logs are not correctly correlated to the matching service. Instead, we see the following:
- tomcat-application
- Logs App 1
- Logs App 2
- Logs ...
- App 1
- Traces
- Errors
- App 2
- Traces
- Errors
- ...

The agent seems to instrument the logback appenders and has no reference to the original application (`event.dataset` is set to `tomcat-application.ApplicationAAppenderDev`, `service.name` is `tomcat-application`). This means, calling https://my-kibana-host:5601/s/my-team/app/apm/services/ApplicationA/logs does not show the application logs.

## Describe the solution you'd like
I would prefer a solution where the `log_sending` logic detects the correct `service.name` the same way as the traces are generated. So I would expect this in Kibana APM UI:
- App 1
- Traces
- Errors
- Logs
- App 2
- Traces
- Errors
- Logs
- ...

## Describe alternatives you've considered
As a workaround we could filter based on the `event.dataset` which seems to be set to the service and logging-appender name. This works for correlating data manually as we have specially named appenders but this will not show the Logs correctly in the Kibana APM UI for the correct service.

## Additional context
Kibana: 8.7.0
APM integration: 8.7.0
APM Java Agent: 1.36.0