On a technical level, how is TikTok being blocked/banned in the US?
Can I still sideload the app to my phone? Is it only being banned from the two big app stores? Is there a penalty for being found in possession of the software on US soil?
Not quite. You don't need any app store at all.
You literally download an APK file from a website or anywhere, then install it directly. Could even be a friend with a thumb drive. Doesn't matter how you get it, it's just a file.
That's got nothing to do with what sideloading means.
And the risks are very overblown.
While it's possible, it's extremely rare.
Mostly because the potent target pool is so small. Bigger potentials for bad guys if they trick app stores into approving trojan horse apps, because everyone thinks app stores are safer.
Public DNS servers hosted in the US will get notified to delist the domain or direct it to an alternate IP. ISPs will get notified to route IP traffic elsewhere.
To be fair, it wouldn't be every ISP that would reroute, just backbone ones. Their routing tables would filter down to regional and last mile networks.