This is the HTML based email you got:
When you click that blue button you're actually navigating to an url that looks like this
https://sg-links.stackoverflow.email/ls/click?upn=[many many chars]. If there is something wrong with the [many many chars] (for example you missed one) you end up with the error message:
You have clicked on an invalid link. Please make sure that you have typed the link correctly. If are copying this link from a mail reader please ensure that you have copied all the lines in the link.
There is no styling with this message, it is plain and boring.
If you get that error message do check the email is complete (notice it has a footer with the address in it), check that the browser you use is able to handle long urls and verify no user scripts, anti virus, firewalls interfere with that url. Most email clients should allow you to right-click on the blue button and copy the link. Try that link in a different browser and/or on a different network to rule out local issues.
The email also does come with a text/plain variant. You might need a decent mail client, like Thunderbird, to be able to switch to that view.
In that plain/text view you'll find an url that looks like
https://meta.stackexchange.com/users/signup-finish?email=[your email and many many charaters]. If there is something wrong with the [many many characters] (for example one went missing) you'll get the message:
The complete signup link you just used is incomplete.
Please try signing up again. If you're still having trouble, contact support.
And that message is shown with styling from the SE site:
If you get that error, double check the email is complete. After the url you'll find the text "Stack Overflow, 110 William Street, 28th floor, New York, NY 10038 <3" so if that is missing or garbled you know something is off. If the link seems complete then try a different browser and/or network and check your antivirus/firewalls or other local software that could interfere with receiving email or sending urls.
There is most likely something wrong on your end, either your internet service provider or mail client doesn't correctly process the received email, causing the link and its security measures to be incomplete. It is not something Stack Exchange can fix for you.