Your Bitcoin balance, as it's shown in Coinbase's UI, is very different from your Bitcoin balance as it's understood by everyone else on the public blockchain; and that too only temporarily. There is nothing that Coinbase can do to permanently alter that supply of Bitcoin. That is what we're talking about.
In contrast, the permanent US Dollar supply can and is permanently altered due to the monetary policy which governs it.
* accept 100 bitcoins for deposit, giving their previous owners "100 claims against exchange X" in return
* sell those 100 bitcoins
* the new owners of those 100 bitcoins can roll down to exchange X and deposit them and accept 100 claims against exchange X in return
* and the exchange can sell those 100 bitcoins once again
Now we've got 100 bitcoins out in the world and 200 claims against exchange X, and each of these 300 items functions identically. They're exposed to all risk of gain or loss on the market, they can be traded, sold, used to purchase stuff... with the only limitation being that not every claim against exchange X can be immediately redeemed. But as long as not every person tries to redeem their claim at the same time, there are no problems.
Exchange X has permanently [until they exit the market] increased the usable supply of bitcoins. Nothing prevents an exchange from having more than 21 million bitcoins on deposit. With large enough exchanges, the banked, usable balance of bitcoins could be 100 million coins. 100 trillion. There is no limit.
You can still transact with Bitcoin using your own wallets against other exchanges/vaults. That's the entire point, Coinbase, like a (lower case "b") doesn't control the supply of Bitcoin, any more than precious metals bullions control the supply of gold despite their issuance of "gold certificates". Like any (lower case "b") bank, Coinbase can engage in bad behavior, and users are placing their trust in Coinbase to not do that. But those users aren't placing their trust in Coinbase to drive the monetary policy of Bitcoin, in the same way that the gold hoarder placing their gold bars in a Swiss vault doesn't really have to worry about the operator of the vault creating gold out of thin air. That fundamental fact drives the collective belief in the value of gold; as well as Bitcoin.
> * the new owners of those 100 bitcoins can roll down to exchange X and deposit them and accept 100 claims against exchange X in return
You can't deposit that purchased BTC at another exchange unless you cash out of Coinbase, because you don't have access to the private key, Coinbase does.
A government absolutely can declare that the correct blockchain is actually this other blockchain, or the only legal algorithm is now a version of their own invention (like the split between Bitcoin and Bitcoin Cash, but enforced by law). If the gov wants inflation, they can replace the algorithm with one that allows inflation, or achieve a similar effect by publishing an official root block every year with a portion of all holdings transferred to the government.
But they probably wouldn't do that, when it's so much easier to simply seize cryptocurrency ("give us your private keys or spend thirty years in prison") or heavily tax it.
None of this has been done because crypto is still a novelty, but if it ever becomes a serious threat to government policy, governments will find ways to deal with it.
And centralizing Bitcoin in exchanges like Coinbase makes this far easier; the government only needs to enforce its policy on Coinbase, and Coinbase in turn will enforce it on customers.
To that point, I found this blog post pretty interesting: https://juraj.bednar.io/en/blog-en/2020/11/12/how-could-regu...
tl;dr: Bitcoin mining is mostly done by businesses, businesses will choose legal compliance over bitcoin-idealism, governments could wrest control of blockchain transactions through legal penalties (e.g. blacklist certain addresses and make it illegal for miners to process transactions from them OR mine blocks based off ones with illegal transactions). Economic incentives will encourage miners too small for enforcement to conform to the regulations. Bitcoin idealists can't do anything about it because their hashrate is puny and no one legit will deal with their forks.
