Actually, yes, sometimes, but it's not always 100% accurate.
In fact, I think that most of my characters originally are based of an emotion I'm feeling when I make them, but sometimes the basis for their creation fades away in time.
For example, Linger (my winged lion), was originally made after a time I'd felt the most loved and almost blessed by a very dear (ex)friend of mine. While the relationship itself may have met a different fate, I would still get onto Linger at times when I felt humbled or shy. However, he eventually became a character that I grew much more comfortable on, and so he was often my usual character that I played.
Nowadays, I'm almost always on Allegra, who is my main fursona. I can't say I need a particular mood to be on her (seeing that I'm on her all the time), but that's because Allegra herself is rather versatile in any situation.
My newest/favorite warrior cat OCs, Lightmare, was made on a rather dark day for me, and there are times when I'm feeling worn down (but still driven by duty to be here for FH) that I'll be on her, since she's a bit of a release from that, simply due to her worn, yet rock-solid personality. However, there are days when I just feel like RPing as her, as well.
Yes and no, but I have indeed done this before.