Does this addition technique have a name?

The individual instance of transferring one digit from one row or column to another in addition or multiplication is called carrying.

The general process of addition using cyclical carrying (or not as the case may be) is called long addition. It takes its place alongside long subtraction, long multiplication and long division. Of these, long division is the most well-known and frequently used term. Together these are sometimes referred to as the long operations. What all of these have in common is the transferring of some sort of remainder or leading digit from one column or row to another.

So, if you ask your maths teacher what you're doing in class today, they'll say "long addition". If someone asks you what you're doing when you put a digit below or above another column, you'll say "I'm carrying the one".

I was taught it as column addition.

Given the webpages listed by Google for this term, I believe it's a British English term.

Google trends supports that:

And it shows that there are almost no searches for "regrouping addition" in the UK: