DO they point to a copy of the same database?

MySQL localhost. Different database different user credentials.

Perhaps a session/cookie prefix can be set in the configuration or set via a random routine. I hadn't had the time to see if two different subdomains would make a difference.

Two cookies are set
PHP Session e.g. PHPSESSID
and
YII_CSRF_TOKEN

Cookie Hostname is www.domain.tld when using subdirectories
Cookie Path is always "/"

When using a subdomain the hostname would always make a difference.

The Cookiepath might has to be set to the subdirectory of the installation to make a difference.

Doc issue : cookie path to / By default is OK.

But doc need an update for this https://manual.limesurvey.org/Optional_settings#Session_settings

@jelo:
You can try:
'session' => array (
'cookieParams' => array(
'path' => '/limesurvey02',
),
),
'request' => array(
'csrfCookie' => array( 'path' => '/limesurvey02' )
),

Alternative (not tested)
'session' => array (
'sessionName' => 'SESSID_206', // For example
),

https://manual.limesurvey.org/General_FAQ#2_limesurvey_instance_on_same_domain

Version 2.06+ Build 150618 released

@mdekker : maybe make another bug report. Because if i understand right:

• Web installer do the job OK, jelo issue are at contraty : the 2 session are diffrent (security is OK) but open one close the other.
• Only command installer : broken : same session from one to other