Please, where are you writing $JOB_NAME value? What job configuration option?

In an execute shell build step and also an execute shell post-build action.

Also, this is unrelated, but I think there are two separate logins for the Jenkins Jira? accounts.jenkins.io and the link on the top right. I had to log in to both of them... Took me a while to figure that out.

Please, give us more details: Jenkins version, execution example, console output, ...

According to logins, accounts.jenkins.io is only to manage your account. You don't need to login twice to use JIRA. Only in https://issues.jenkins-ci.org.

Job config attached. config.xml

Jenkins ver. 1.642.4

Started by upstream project "XIAN_TENG_TEST_JOB" build number 1
originally caused by:
 Started by user xian.teng
[EnvInject] - Loading node environment variables.
Building remotely on ubuntu-1204-03-64 (qa_runner admin_linux build_docs 12.04 amd64-Ubuntu Ubuntu amd64-Ubuntu-12.04 Ubuntu-12.04 amd64) in workspace /jenkins_data/jenkins/workspace/XIAN_TENG_TEST_JOB/parameter1_axis/value1/parameter2_axis/value2
[value2] $ /bin/sh -xe /tmp/hudson5641186992722460038.sh
+ echo build step
build step
+ echo value1
value1
+ echo value2
value2
+ echo XIAN_TENG_TEST_JOB/parameter1_axis=value1,parameter2_axis=value2
XIAN_TENG_TEST_JOB/parameter1_axis=value1,parameter2_axis=value2
Finished: SUCCESS

This looks like behavior as intended. For Matrix projects, the individual sub-projects have names according to parameter names, which is independent from the parent name (the name of the matrix project). See https://github.com/jenkinsci/matrix-project-plugin/blob/master/src/main/java/hudson/matrix/MatrixConfiguration.java#L97

In your environment JOB_NAME is being set to the full path to the job. In XIAN_TENG_TEST_JOB/parameter1_axis=value1,parameter2_axis=value2, XIAN_TENG_TEST_JOB refers to the parent of your job (like with folders), and parameter1_axis=value1,parameter2_axis=value2 is the actual job name.

In newer versions of Jenkins, the environment will also contain JOB_BASE_NAME, which in your case would just be parameter1_axis=value1,parameter2_axis=value2. As far as I know, the parent name is not injected into the environment, though that might be a decent feature ask to put in (at least for matrix jobs?)

From what I see it is a duplicate of JENKINS-28176 

The documentation is just outdated, but actually the full name with parameters is valid for Matrix projects