"Floyd Rose Licensed" basically just means that the manufacturer has paid a licensing fee to the Floyd Rose company so as to not infringe upon Floyd Rose's patents. The trem could be made of stale chewing gum; as long as it is based upon the Floyd Rose design, it must be licensed.
So, chances are, if you bought a dirt-cheap trem on eBay, it's going to behave like it was dirt cheap. The fabricator is still what you need to go by in terms of your expectation of quality.
The Floyd Rose company themselves of course makes a full line of tremolos; their website also contains a page about how to tell a genuine trem from a licensed one.
I generally wouldn't trust a "Floyd Rose Licensed" trem that didn't have a known brand associated with it. For example, Ibanez is an established guitar manufacturer, and many of their Edge tremolo designs are or were subject to the Floyd Rose patents, and as such were Floyd Rose Licensed tremolos. I personally have an ESP guitar with a Licensed tremolo that I've never had any problems with, though it appears that now they actually throw genuine Floyd Rose tremolos on their current models.
It's also worth noting that the original Floyd Rose patent expired in 2003, so tremolos based on the original design don't need to say "Floyd Rose Licensed" on them in the first place.
So at the end of the day "Floyd Rose Licensed" is effectively meaningless in terms of quality. If you need something reliable, you should probably make an investment in a trem from a reputable parts manufacturer.