This will be a mainly negative answer - not in the ad hominem sense, but I mainly discuss how a serial number often fails to identify a bicycle, especially if someone presents a serial number and no other information (especially manufacturer of the bike or at the very least a photo), and asks us to identify the bicycle. The executive summary is that a serial number alone cannot identify a bicycle. Perhaps we can point people to this answer when they present a serial number.
A previous answer on this question says:
Just like a VIN number on a car, your serial number helps authorities determine who the lost or stolen bike belongs too.
Bicycle serial numbers are not exactly like Vehicle Identification Numbers (VINs). A VIN is unique to each car, and there is at least one global database of VINs. As a consumer, you can buy an automobile history report if you present the VIN of a prospective purchase to a commercial provider. At least in the US, you can check some limited info for free at the National Insurance Crime Bureau, which is a non-governmental non-profit organization.
Note the bit about insurance. Vehicles are expensive, and repairing them is expensive. Because of their mass and potential velocity, they can cause significant damage to humans and other vehicles. Without getting too deep into the history of automobile insurance, this was part of the impetus for this type of insurance to even exist and to become mandatory in, at the very least, most US states. (And if you are allowed to drive uninsured, you will still be liable for damage that you cause, but that's secondary to the topic.) Also, there is a robust secondary market for motor vehicles, which is another impetus for a global identification number.
The Wikipedia page on VINs doesn't say why they were created. I'd hypothesize that a major part of the impetus was the need for insurance (and thus, the need to identify the unique vehicle under contract and to identify parties in an accident) and the issue of theft (i.e. identify your car if recovered). Whatever the case, there were some collective action problems in even creating VIN formats, so the International Organization for Standardization had to step in to propose formats, and national governments had to create legislation to mandate that these formats be used.
Bicycles are often used as recreational vehicles and as secondary transport vehicles. They can certainly injure pedestrians and other bicycles, but the potential scope of injury is less. Thus, there has been less impetus to create a national mandatory database of identification numbers for bicycles. They tend to be much less expensive than cars - it is possible to build top end bicycles at a comparable price to many cars, but exceedingly few people spend this amount. Bicycle insurance is a niche product, at least in the US (this covers events not covered by one's homeowner's or renter's insurance), and not many people buy it, even if they have expensive bicycles. One example of a bicycle insurance provider at the time of writing is Velosurance. They most likely are able to identify their covered bicycles through what serial numbers are provided, as well as other identifying information (e.g. photograph the bicycle).
Indeed, to my knowledge, there is no national database of bicycle registration numbers that all manufacturers must automatically submit data to. There are some voluntary bicycle registration databases, e.g. the National Bike Registry, Bike Index, and the International Bike Registry. Their existence and utility are discussed in this Stack Exchange Q&A. However, you have to know they exist, and if you check a bike there, the original owner would have had to know about the bicycle and register it there. Thus, these systems likely cover only a small minority of bicycles. Also, that Q&A mentioned that there are some bicycle registration databases that are exclusive to US law enforcement, but again, they're voluntary, and the police department has to elect to use them and to keep consistently inputting data (presumably on recovered bicycles, but the answer wasn't clear on that).
Even if you know the bicycle's manufacturer, the serial number likely provides very little information to the general public. For prominent manufacturers, it's possible that other riders have deduced the format of the serial number, e.g. some information can be gleaned through the first two characters of Schwinn serial numbers, at least for some manufacture dates. But do note that that site doesn't say that you can identify a Schwinn model from the serial number, and it says that the formats varied by place of manufacturer. I think that some Serotta aficionados can deduce the frame model from the serial number. However, how many people have even heard of Serotta? It was an eponymously named company that did high-end steel and titanium frames, mix of stock and custom. The fact that some people can identify Serotta models and years of manufacture given a serial number is very unrepresentative. Also, some Colnagos may be identifiable through their serial numbers, but Chuck Schmidt of the site Velo-Retro.com had this to say:
Serial numbering only started for Colnago on USA-bound bikes when the CPSC started requiring tracking, etc. West Coast Cycles was importing Colnago at the time and the serial numbers started for their product. There is still no way to find manufacturing date by the numbers, except that they are 1979 or later…
If you present a manufacturer with a serial number, then they can often identify your bike's make and model. I am not sure if they can always do so, as I have no experience nor empirical evidence. If they used inconsistent serial number formats in the past, the task would be harder. Also, corporate ownership can disrupt record keeping and retention of internal knowledge. For example, Schwinn Bicycles was independent from 1895 to, I believe, 1992 when they declared bankruptcy. They were purchased by a series of investment funds, merged with GT, went bankrupt again, and finally purchased by a firm that supplies - again, not meant as ad homimen - low-end department store bicycles. If you called the current Schwinn Bicycles, the current corporation's staff might or might know the historical serial number formats or be able to look them up. Or they might decide that it's not worth identifying historical Schwinns for second (or third, fourth, etc) hand owners. Or if you are asking for specific information on a single bicycle, as opposed to just year of manufacture and model, then this might call for detailed individual records, and these might have been lost in change of ownership, or might have been pen and paper originally and not converted to machine-readable format (i.e. if you scanned PDFs of the documents, they might not be searchable by the computer).
It is true that you can register your bicycle with the police. This may help prove your ownership if it's stolen and later recovered - if the police recover it, if they can be bothered to consult their records, and if they kept adequate records to begin with. Remember that many bicycles are stolen, and their commercial value is much less than cars. @paparazzo's answer said local police. To elaborate more: if you register your bicycle with your local police force, and if that same force recovers your bicycle and they can be bothered to check the serial number against their registry, this could lead to a notification. Police may genuinely have more pressing cases to clear, e.g. violent crimes. Also, for higher-end bikes, my understanding is that many criminals will steal in one location and then fence the bike in another; it's possible that the police forces in both locations are using the same bike database, but not a guarantee.