Those "Analyze Board" links are coded to open a popup window using JavaScript. Your desktop browser (using an extension) is overriding this default behaviour it would seem, in order to open in a new tab instead.
Those links are not programmed in a particularly accessible way unfortunately - if you don't have JavaScript enabled you get nothing. Under Google Chrome on the desktop for instance I cannot avoid opening a popup window, it simply doesn't work if I choose to open the link in a new tab (I get a blank page).
Having said that, using the default browser on Android 4.0, does open a new tab when those links are clicked - I don't get a floating popup window.
I would have said that the problem is a combination of the website (not being accessible) and of the default browser on your mobile that is unfortunately unable to deal with this situation. Can popup windows be controlled in the browser settings? In some cases you could disable JavaScript on the mobile to get around this, but on this website if you did that you would get nothing!
EDIT: You mention that you don't have much room on your device to install an additional (full blown) browser. Have you considered perhaps Opera Mini (858KB) or Dolphin Mini (1,014KB) - these have a reduced storage footprint.