Using Raku (formerly known as Perl_6)
raku -ne '.split(/ <[/=]> /).[2,4,7].put;'
Sample Input:
/logs/tc0001/tomcat/tomcat7.1/conf/catalina.properties:app.env.server.name = demo.example.com
/logs/tc0001/tomcat/tomcat7.2/conf/catalina.properties:app.env.server.name = quest.example.com
/logs/tc0001/tomcat/tomcat7.5/conf/catalina.properties:app.env.server.name = www.example.com
Sample Output:
tc0001 tomcat7.1  demo.example.com
tc0001 tomcat7.2  quest.example.com
tc0001 tomcat7.5  www.example.com
Above is a solution coded in Raku, a member of the Perl-family of programming languages. Briefly, input in read linewise with the -ne (linewise, non-autoprinting) commandline flags. Lines are split on a regex which consists of a custom character class (/=) created with the <[ ]> operator. Elements [2,4,7] are then put to give the results above.
Of course, the above is a 'bare-bones' implementation, and Raku being a Perl-family language, TMTOWTDI applies. So lines can be split on literal characters separated by a | "OR" operator. Element numbering (which is zero-indexed in both Perl and Raku) can be tightened up adding the :skip-empty adverb to the split routine. Whitespace can be trim-med away from each element (using map), and the desired elements (now [1,3,6]) are join-ed with \t tabs, giving the following result:
raku -ne '.split(/ "/" | "=" /, :skip-empty).map(*.trim).[1,3,6].join("\t").put;' file
tc0001  tomcat7.1   demo.example.com
tc0001  tomcat7.2   quest.example.com
tc0001  tomcat7.5   www.example.com
https://raku.org