Exclaves and Enclaves
@Ash has hit on a very important issue: you can't just make geography go away. Contiguous borders are very convenient. They provide easy travel and communication between your people, and a minimal border to control and defend.
What you're proposing is that everyone lives in enclaves and exclaves: territories surrounded by other territories. If they want to get anything physical done they must negotiate with their neighbors.
For example, my house has hookups for roads, water, sewer, gas, and electrical. These will necessarily have to pass over and under my neighbor's property who may be in other Commonwealths. I need physical access to and from my property to get supplies and visit friends. What if my neighbor doesn't want that? I'm effectively blockaded and my Commonwealth will have to negotiate peacefully or with force to keep me alive.
Hopefully the utility companies will do that negotiating for me, but they will rapidly tire of having to negotiate with thousands of little Commonwealths each with their own political structures and wants. It's not an environment conducive to business. And when they get it all sorted, maybe somebody decides to change to a Commonwealth they don't have a treaty for and they have to start the process all over again.
Similarly, how does a geographically fractured Commonwealth physically protect its members? With potentially every citizen in their own exclave its surface area is huge! They can't post a guard in each exclave. When someone breaks into my house, how far away will the police be? Will they have to negotiate with my neighbors to come to me?
Finally there is public safety. My neighbor's house catching on fire puts my house in danger. My neighbor burning tires puts my health in danger. My neighbor shooting guns puts my life in danger. This is why we have building codes and environmental laws, we all breath the same air and drink the same water. What happens when I watch my neighbor in a different Commonwealth do some really dodgy wiring, smoke in bed, burn their trash, and poop upstream from me?
This means Commonwealths must negotiate treaties with other Commonwealths in order to function. Not just their immediate neighbors, but they must negotiate a contiguous network for utilities and transportation. There's two ways this could happen...
Let the Corporations handle it. This is kind of like what is done today in the US where governments use their power of Eminent Domain to seize land (with compensation) for the public good to build roads, utilities, parks, etc. Normally this land must be used by the government alone for the public good, but delegation to a third party is also possible. For example, governments often seize land and hand it to utility companies and railroads to run their lines. For Commonwealths this might also include police, medical, and firefighting.
The danger is this would, effectively, hand much of the power of a Federal government over to Corporations. Similar to the issue of multi-national Corporations today, but on a much, much, much larger scale. Successful Corporations would be able to cross Commonwealth borders and scale far, far, far larger than any single Commonwealth.
On the upside, this would make them more efficient. On the downside, this would make them larger and more powerful than any Commonwealth. It would give the Corporations coercive control with no regulatory oversight. Rather than fight each other for Commonwealth contracts, the few largest corporations would divvy up the market and form a coercive monopoly to maximize profit for themselves. A modern example is the US Internet market where 2/3 of the population has 2 or fewer options.
If you think End User License Agreements are bad now, wait until there's no government oversight. Your Internet company would be free to spy on everything you do. Don't like it? No Internet. One might think "I'll just use encryption and darknets", nope; if you use anything but the approved encryption (complete with Corporate backdoor) you're banned. Net Neutrality goes right out the window, you're stuck only contacting the Corporate approved Internet services that they have profit sharing deals with.
How about water? Don't want to pay their prices, your Commonwealth dries up. At the extreme end, you are invaded by a Corporation's overwhelming private security firm.
If they don't want a cyberpunk corporate dystopia, Commonwealths would form a Confederation to function. Rather than simply negotiating with each other, Commonwealths would join a Confederation (there need not be only one). They'd delegate some of their powers to it, and gain some power to influence the decisions of the Confederation. Exactly how influence is distributed is up to the Confederation.
For example, members of a Confederation might delegate their right to border control to the Confederation resulting in free passage for their members between all other member's territory like the Schengen Area. They might agree to a single currency, like the Eurozone, and delegate some of their monetary policy to a central bank. They might negotiate a common security and defense policy. Water and power security. A transportation system...
The Confederation also acts like a co-op or trade union. It leverages the collective bargaining power of the Commonwealths against other Confederations, Commonwealths, and Corporations. This multilateral bargaining power gives each Commonwealth power far beyond its own small power. The whole is greater than the sum of its parts.
Yes, this is starting to sound like the European Union for a good reason. The EU is really a collection of treaties between its members. CGP Grey does a good breakdown of its structure.
Commonwealths, Confederations, and Corporations
It's not either Confederations or Corporations. It's both. Multinational Corporations will form and exploit the divisions between the Commonwealths. Commonwealths will band together into Confederations to increase their collective utility, bargaining, and defensive power against other Commonwealths, Confederations, and reign in the Corporations.
Beyond just property and defense, it is in each Commonwealth's long term best interest to ensure their neighbors are also stable and healthy. Disease, fire, pollution, bullets, and unrest know no borders.
Commonwealth members have two choices in the long term.
They can go it alone and live in a world of armed, walled enclaves trying to ignore the problems of their neighbors. To get services they'll hand more and more power over to the Corporations. Like a modern gated community, everything will be fine so long as their wealth holds out.
They can band together into Confederations for collective security and mutual benefit. All for one and one for all. Each Commonwealth contributes to the Confederation and the Confederation supports each Commonwealth. The Confederation can have their own utilities and services, or can bargain collectively with Corporations reducing the threat of Corporate take over.
The third option, have a single Commonwealth that is large enough and resource rich enough that it can go it alone, is unstable. It is effectively a Confederation with no governing body. Any single member can decide to switch to a new Confederation possibly throwing the whole situation into chaos.