Add Fleet agent.id to Agent monitoring data (#26548)

This is a follow up to #26394 which set the agent.id field to the Fleet agent ID for
for integrations that are run by Agent, but that didn't cover the Filebeat and Metricbeat
processes that are executed by Agent for monitoring itself. This covers those processes.

Relates #26394

(cherry picked from commit 7b66597)

x-pack/elastic-agent/CHANGELOG.next.asciidoc

@@ -116,5 +116,7 @@
 - Use `filestream` input for internal log collection. {pull}25660[25660]
 - Enable agent to send custom headers to kibana/ES {pull}26275[26275]
 - Set `agent.id` to the Fleet Agent ID in events published from inputs backed by Beats. {issue}21121[21121] {pull}26394[26394]
+- Set `agent.id` to the Fleet Agent ID in events published from inputs backed by Beats. {issue}21121[21121] {pull}26394[26394] {pull}26548[26548]
+- Add proxy support to artifact downloader and communication with fleet server. {pull}25219[25219]
 - Enable configuring monitoring namespace {issue}26439[26439]
 - Communicate with Fleet Server over HTTP2. {pull}26474[26474]

x-pack/elastic-agent/pkg/agent/operation/monitoring.go

@@ -339,6 +339,16 @@ func (o *Operator) getMonitoringFilebeatConfig(outputType string, output interfa
 		"output": map[string]interface{}{
 			outputType: output,
 		},
+		"processors": []map[string]interface{}{
+			{
+				"add_fields": map[string]interface{}{
+					"target": "agent",
+					"fields": map[string]interface{}{
+						"id": o.agentInfo.AgentID(),
+					},
+				},
+			},
+		},
 	}
 
 	o.logger.Debugf("monitoring configuration generated for filebeat: %v", result)
@@ -563,6 +573,16 @@ func (o *Operator) getMonitoringMetricbeatConfig(outputType string, output inter
 		"output": map[string]interface{}{
 			outputType: output,
 		},
+		"processors": []map[string]interface{}{
+			{
+				"add_fields": map[string]interface{}{
+					"target": "agent",
+					"fields": map[string]interface{}{
+						"id": o.agentInfo.AgentID(),
+					},
+				},
+			},
+		},
 	}
 
 	o.logger.Debugf("monitoring configuration generated for metricbeat: %v", result)