Aaaaand they're also similar in all the other ways I just mentioned, a couple of which are quite remarkable.
The only similarities in the things you mentioned are covered in what I said.
1. The 'New California Republic and the Legion are quite similar to the Boulder Free Zone and Las Vegas factions from The Stand.
Not really, no. They have almost nothing in common.
2. Mr. House from NV seems (from what I've heard) to resemble the character of Randall Flagg, to a degree.
Along with thousands of other antagonists. I would not be surprised, however, if he's not inspired by Randall in some way. That doesn't make New Vegas like The Stand in any remarkable way.
3. The use of crucifixion as punishment, involving a crossed pair of telephone poles, in both stories.
It's a real punishment that has existed for thousands of years, hardly remarkable.
4. Both take place after most human life has been exterminated, nuclear war being the cause in Fallout, the "Captain Trips" superflu being the cause in The Stand.
Like I said, they're post-apocalyptic. If having the same general theme is remarkable then every high fantasy game, movie, or show is remarkably similar to each other as well.
5. The general location of the stories are both set in the area of Las Vegas and Nevada, as well as Colorado in The Stand.
They share one location, Las Vegas.
If you consider sharing one location and a very broad theme remarkable then OK. Using this logic I could find that just about any entertainment product is remarkably similar to any other. After all I'm sure I can find an evil mastermind in Spiderman, and they both involve a city, and they're both in the United States, and they both have mutants. They similarities there are remarkable.