"Wankese" specifically refers to the meandering, pointless, angst-filled voiceovers provided by Tightass in Final Fantasy X (and then by Yuna, striving to become her boyfriend in every way, in Final Fantasy X-2). It's generally used elsewhere to refer to any navel-gazing voiceover bullshit by characters we dislike.
Examples
Now Tightass slips back into some bad Wankese: "I had made it out of the frying pan and into the freezer. I thought I was going to die in this place." Nope, not yet, Tightass. Sorry to disappoint me. - Final Fantasy X Part 1