"Objective IELTS" is a private company and not any kind of official testing agency. They probably give good advice, but that does not mean they are always correct.
However, in this case the important information is that the IELTS may mark incorrect even minor mistakes such as where adverbs are placed in a sentence. While the example sentence is grammatical, it would be better to place the adverb closer to the verb it modifies.
The problem I have with this sentence is that even the "correct" answer is awkward. I don't like ending with "since around 1920", because it's both unnecessarily confusing and unnecessarily vague.
It's unnecessarily confusing because, when added at this point in a sentence, "since" frequently means "because". You should avoid writing sentences that make the reader unsure about the meaning of ambiguous terms.
It's unnecessarily vague because 1920 was quite some time ago, so unless the date relates to some important information, it's hardly necessary to qualify with "around 1920". No one cares if the date is not exact.
Overall, it's not clear what the sentence intends to say. I understand what it means by itself, but it doesn't seem to fit elegantly into any larger context. Here's one way I would have written this sentence, in order to better lead into a discussion of these changes in customs:
It seems that Japanese customs have changed, little by little, from what they were in 1920.
I understand this may be of little help for the official examination, but it irritates me when a student is asked to "fix" a sentence that is fundamentally broken.