The key to solve this is to know which instance of selenium you are refering to when you call open_browser and that can be controlled by being explicit refering to the Selenium remote RC client_web.open_browser or by using Keyword Set Library Search Order.
*** Settings ***
Library SeleniumLibrary 120 ${CLIENT_IP} ${SELENIUM_SERVER_PORT} WITH NAME client_web
*** Variables ***
${MANAGER_BROWSER} Firefox
${BROWSER} Firefox
*** Test Cases ***
Verify Two Browsers Using Search Order
[Setup] Test Case Setup
[Tags] noncritical
Set Library Search Order SeleniumLibrary
${wb_index} = Open Browser ${DUT} ${MANAGER_BROWSER}
Set Library Search Order client_web
${wb_index} = Open Browser ${facebook} ${BROWSER}
Maximize Browser Window
Select Window main
Wait Until Page Contains ${page text} 10s
Log browser-index:${wb_index}
Comment Set suite variable Manager_Browser with call to open specific browser
[Teardown] Local Test Case Teardown