Tomcat 8.5 context path not working correctly

I don't understand why you are having problems, but I've created the web project using NetBeans 8.2 (File > New Project... > Java Web > Web Application), and successfully changed the context, so perhaps if I give details of my project you can identify where there's a crucial difference.

1 cat_tiger\web\META-INF\context.xml

My file looks identical to yours:

<?xml version="1.0" encoding="UTF-8"?>
<Context copyXML="true" docBase="cat_tiger" path="/cat/lion"/>

[2] Run output

Select the project node, right click and select Run from the popup menu to run on Tomcat 8.5:

ant -f D:\\NB82\\cat_tiger -Dnb.internal.action.name=run -Ddirectory.deployment.supported=true -DforceRedeploy=false -Dnb.wait.for.caches=true -Dbrowser.context=D:\\NB82\\cat_tiger -Duser.properties.file=C:\\Users\\johndoe\\AppData\\Roaming\\NetBeans\\8.2\\build.properties run
init:
deps-module-jar:
deps-ear-jar:
deps-jar:
library-inclusion-in-archive:
library-inclusion-in-manifest:
compile:
compile-jsps:
In-place deployment at D:\NB82\cat_tiger\build\web
Deployment is in progress...
deploy?config=file%3A%2FC%3A%2FUsers%2Fjohndoe%2FAppData%2FLocal%2FTemp%2Fcontext7953615149857268018.xml&path=/cat/lion
OK - Deployed application at context path [/cat/lion]
Start is in progress...
start?path=/cat/lion
OK - Started application at context path [/cat/lion]
run-deploy:
Browsing: http://localhost:8080/cat/lion
run-display-browser:
run:
BUILD SUCCESSFUL (total time: 0 seconds)

Actually, you probably don't need to run the application to see the problem; just select Deploy instead of Run from the popup menu. This is the output I get from Deploy:

ant -f D:\\NB82\\cat_tiger -Dnb.internal.action.name=redeploy -Ddirectory.deployment.supported=true -DforceRedeploy=true -Dnb.wait.for.caches=true -Dbrowser.context=D:\\NB82\\cat_tiger -Duser.properties.file=C:\\Users\\johndoe\\AppData\\Roaming\\NetBeans\\8.2\\build.properties run-deploy
init:
deps-module-jar:
deps-ear-jar:
deps-jar:
library-inclusion-in-archive:
library-inclusion-in-manifest:
compile:
compile-jsps:
Undeploying ...
undeploy?path=/cat_tiger
OK - Undeployed application at context path [/cat_tiger]
In-place deployment at D:\NB82\cat_tiger\build\web
Deployment is in progress...
deploy?config=file%3A%2FC%3A%2FUsers%2Fjohndoe%2FAppData%2FLocal%2FTemp%2Fcontext5063723197082921373.xml&path=/cat/lion
OK - Deployed application at context path [/cat/lion]
Start is in progress...
start?path=/cat/lion
OK - Started application at context path [/cat/lion]
run-deploy:
BUILD SUCCESSFUL (total time: 0 seconds)

[3] Tomcat log

Here are the deployment details in the Tomcat log, where you can see that my deployment shows the context correctly, whereas yours does not:

05-Sep-2018 23:09:09.321 INFO [http-nio-8080-exec-6] org.apache.catalina.startup.HostConfig.deployDescriptor Deploying configuration descriptor [C:\apache-tomcat-8.5.20\conf\Catalina\localhost\cat#lion.xml]
05-Sep-2018 23:09:09.334 INFO [http-nio-8080-exec-6] org.apache.catalina.startup.HostConfig.deployDescriptor Deployment of configuration descriptor [C:\apache-tomcat-8.5.20\conf\Catalina\localhost\cat#lion.xml] has finished in [13] ms
05-Sep-2018 23:09:09.338 INFO [http-nio-8080-exec-5] org.apache.catalina.util.LifecycleBase.start The start() method was called on component [StandardEngine[Catalina].StandardHost[localhost].StandardContext[/cat/lion]] after start() had already been called. The second call will be ignored.

[4] C:\apache-tomcat-8.5.20\conf\Catalina\localhost\cat#lion.xml

This is the file Tomcat created when deploying the application:

<?xml version="1.0" encoding="UTF-8"?>
<Context copyXML="true" docBase="D:\NB82\cat_tiger\build\web" path="/cat/lion"/>

Note that docBase contains an absolute path.

[5] server.xml

Here's the entire content. I'm using Tomcat 8.5. Note that the <host> element is slightly different to yours, but when I added copyXML="true" deployXML="true" to the <host> element (so it looked like yours) everything continued to work fine:

<?xml version="1.0" encoding="UTF-8"?>
<Server port="8005" shutdown="SHUTDOWN">
    <Listener className="org.apache.catalina.startup.VersionLoggerListener"/>
    <Listener SSLEngine="on" className="org.apache.catalina.core.AprLifecycleListener"/>
    <Listener className="org.apache.catalina.core.JreMemoryLeakPreventionListener"/>
    <Listener className="org.apache.catalina.mbeans.GlobalResourcesLifecycleListener"/>
    <Listener className="org.apache.catalina.core.ThreadLocalLeakPreventionListener"/>
    <GlobalNamingResources>
        <Resource auth="Container" description="User database that can be updated and saved" factory="org.apache.catalina.users.MemoryUserDatabaseFactory" name="UserDatabase" pathname="conf/tomcat-users.xml" type="org.apache.catalina.UserDatabase"/>
    </GlobalNamingResources>
    <Service name="Catalina">
        <Connector connectionTimeout="20000" port="8080" protocol="HTTP/1.1" redirectPort="8443" relaxedQueryChars="[]|{}^+\`&quot;&lt;&gt;"/>
        <Connector port="8009" protocol="AJP/1.3" redirectPort="8443"/>
        <Engine defaultHost="localhost" name="Catalina">
            <Realm className="org.apache.catalina.realm.LockOutRealm">
                <Realm className="org.apache.catalina.realm.UserDatabaseRealm" resourceName="UserDatabase"/>
            </Realm>
            <Host appBase="webapps" autoDeploy="true" name="localhost" unpackWARs="true">
                <Valve className="org.apache.catalina.valves.AccessLogValve" directory="logs" pattern="%h %l %u %t &quot;%r&quot; %s %b" prefix="localhost_access_log" suffix=".txt"/>
                <Context docBase="C:\apache-tomcat-8.5.20\wtpwebapps\PlusServlet" path="/PlusServlet" reloadable="true" source="org.eclipse.jst.jee.server:PlusServlet"/>
            </Host>
        </Engine>
    </Service>
</Server>

Updated 9/6/2018:

Ignore the answer given above! Even though it happened to work for my configuration, the approach is explicitly disallowed in the Tomcat documentation.

Specifically, path should not be specified within the <Context> of a context.xml file placed within the application's META-INF directory. From the path description in the Common Attributes section of the Tomcat 8.5 documentation for the Context Container:

This attribute must only be used when statically defining a Context in server.xml. In all other circumstances, the path will be inferred from the filenames used for either the .xml context file or the docBase.

Even when statically defining a Context in server.xml, this attribute must not be set unless either the docBase is not located under the Host's appBase or both deployOnStartup and autoDeploy are false. If this rule is not followed, double deployment is likely to result.