Or is it both? If it is American English, what would be the British English equivalent, or vice versa.
"I really like this dress" as in "I very much like this dress"
There's not really (put intended) a difference in how often it occurs, in any case not significant enough to label either usage as American or British English.
Let's look at the usage of "I really like" and "I very much like"; that should eliminate most possibilities mentioned by @Laurel in the comments.
aren't that much different; you do see that 'really' gained a lot of traction in the last 60 years.