JIRA installs its service as "JIRA Atlassian" therefore by installing the second instance or the following instances, the other instances will try to install their own service as "JIRA Atlassian" too, which make conflict with the first instance or previous instances.

Below log can be seen in the commons-daemon.2015-03-20.log at
$JIRA_INSTALL\logs

[2015-03-20 09:11:23] [info]  [ 9140] Commons Daemon procrun (1.0.15.0 32-bit) started
[2015-03-20 09:11:23] [info]  [ 9140] Service JIRA200315090923 name Atlassian JIRA
[2015-03-20 09:11:23] [error] [ 9140] The name is already in use as either a service name or a service display name.
[2015-03-20 09:11:23] [error] [ 9140] Failed installing 'JIRA200315090923' service
[2015-03-20 09:11:23] [error] [ 9140] The name is already in use as either a service name or a service display name.
[2015-03-20 09:11:23] [error] [ 9140] Commons Daemon procrun failed with exit value: 8 (Failed to install service)
[2015-03-20 09:11:23] [error] [ 9140] The name is already in use as either a service name or a service display name.

Workaround

To rename the older instance's service name.

1. Try to find the older Service Name, for example; JIRA101214115546
   You can use the following command to get the Service Name of the old Windows Service (usually named as "Atlassian JIRA"):

   SC GetKeyName "Atlassian JIRA"
   

   Result:

   [SC] GetServiceKeyName SUCCESS
   Name = JIRA101214115546
   

2. Replace the ID in the following command and preferred new name for older service instead of "Atlassian JIRA"

   SC Config "JIRA101214115546" DisplayName= "Atlassian JIRA 6.3.14"
   

   The old Windows Service has been renamed.

3. Install a new Windows Service by following the steps in Manually setting up JIRA to run as a service.