The court system and government should treat people as innocent until proven guilty. Private citizens have a lower responsibility.
At no point did the original poster claim that man stole his laptop - he merely stated that his laptop was stolen and posted pictures of the man currently possessing the stolen laptop. Other people assumed this man stole the laptop, which may or may not be accurate.
The man is in possession of stolen property. At some point we need to balance the owner's rights to retrieve stolen property against the right's of a man possessing stolen property to retain complete anonymity.
If elected officials do not enforce the law, the burden falls onto private citizens to do so. This is the common law with thousands of years of precedent.
2. Person A no longer has a laptop. Person A did not sell or give away this laptop. The laptop was stolen.
3. Person B now has the missing laptop.
Conclusion: Person B is in possession of a stolen laptop.
If you're not doing that, then no, it needn't be.
This guy also has every right to collect and publicize any information collected by the laptop which will help him locate it. He's titled it "This guy has my MacBook", not stole, and he hasn't posted anything irresponsibly that I've seen.
He has every right to collect the guy's personal information, too, but that doesn't mean he can use his credit card or publish his social security number.
Conclusion?
Now onto a subjective matter. A bicycle is a highly personal item for many people. It may not involve identity theft, but for some people it involves a loss of freedom and mobility that can't be replaced easily. My current bicycle has been highly personalized for my use. Having it stolen would be like having a bit of my soul stolen.
If you don't feel the same way about your bike, I quite understand.
UPDATE:
You said that "The dude lost his right to privacy when he stole a laptop." Did you notice what you just did? You convicted someone of an offence without trying them in a court of law. If we were in a jurisdiction where murder is a capital offence, you could use exactly the same logic for lynching and hanging a man without bothering to try him. He lost his right to life when he murdered, and we know he murdered, so fetch a rope and a horse.
Now some (strawman alert) might say that the evidence is overwhelming. Well, where do we draw the line? What's the rule for "We don't need a trial?" Pictures on the Internet? Eyewitness accounts?? Confessions? All of these have been proved unreliable in the past, which is why we have trials.