twittertwitter-search
I've recently created a Twitter account and my trends are listed like this: 
Yet for other members of my household their list looks like this:
I've tried changing the settings but the results are still the same, I don't even have the same trends showing as them even though we're set to the same location. I would appreciate any help.
Yes, each Twitter Account is assigned a user_id.
Taken from the Twitter API Documentation
user_id
The ID of the user for whom to return results for. Helpful for disambiguating when a valid user ID is also a valid screen name.
It's also useful to know that you can get the user_id by displaying a users tweet via the /statuses/show/{tweetId} API Call
This will display results similar to this:
As you can see the user_id is provided along with the individual tweet information.
The {tweetId} can be found by clicking on the time stamp of any tweet. The {tweetId} will be long number at the end of the URL.
I'm having the exact same problem - goes through the approval process fine, but nothing appears on the page. As you say, profile is working fine, just pages. I tried different accounts and browsers, as well as a few tricks, same result. I can only assume it's a bug as the whole Twitter Facebook thing has been shifty this month. One workaround is to use the facebook app Selective Tweets. This will post to pages; however unlike the native twitter app, each tweet you want to go to the facebook page you must end with #fb, hence the whole "Selective" thing. But at least they show up (and the #fb tag won't show on the page).
Best Answer
As @Andra indicated, trends are based on more than just your location. From the Twitter trends FAQs: