Why does my phone say "Welcome to Twitter"?

Sounds like a trip to VZ. You must have a SMS phone which can be used for Twitter.

40404 is Twitter’s text address, so unless that is somehow spoofed (faked) on the texts that you’re getting (and I wouldn’t know anything more about that than that it ‘might’ be possible), that’s Twitter’s number in the text world. (I do have a Twitter account and I do get occasional notifications via phone, so I know that, anyway.)

It might be possible to put that on a “block” list (blacklist) of text numbers not to contact you. I generally do that via MrNumber, a free app that lets me look up the names that go with unknown telephone contacts, and what other people are saying about “calls from that number”, so I use MrNumber to block telephone contacts. Whether that would work with texts or not, I do not know.

On the other hand, you might also consider contacting Twitter in a businesslike and cordial way and informing them that you have no interest in opening an account or pursuing any kind of relationship with them, and ask them to make the pro-active move to remove you from their contact list.

I wonder if someone either accidentally used your number (my fat fingers have made me input my number wrong) or like you, didn’t want to give a real number and instead of zeros just used a random number that happened to be yours.

I suspect that @Judi‘s answer is the right one.

I had considered that as I composed my response, but I (mistakenly) believed that this was a recurring message on your phone, which would not be an SOP for new account creation at a reputable operation. But you didn’t say that in the OP, so now I believe that @Judi is correct: This was the normal, one-time-only “Welcome to Twitter!” message that any new user would get upon registration, and it probably also includes “next steps” to take to “verify your phone number and user name”, etc. and then (probably) a link to download the app to the mobile device, etc.

So, assuming that to be the case:
1) that the message really IS from 40404, and is therefore a “real” Twitter message / contact;
2) that it has only been sent one time (because Twitter wouldn’t hound you to register if you fail to respond to the initial contact, which they believe you to have made);
3) that it otherwise appears to be genuine,
4) then Twitter believes that they’re simply signing up a new user, per their normal process.

In that case, I think if you simply delete the message then that will be the end of it.

(The reason that you’re getting the message is probably what she said, too: Someone entered your number by mistake instead of their number – an honest error – and Twitter is behaving normally to what they believe is a valid “new account” request. This could also occur if someone did that maliciously – or as some kind of misguided prank – in which case it might be possible for the event to recur until the person tires of the prank, or you take other steps to correct it.)

^^ if there’s confirmation link then there should be another link which should say you’re not intended recipient which happens during email confirmation. Not sure about message though!