Learn English – Use of the suffix -ish

This -ish suffix itself isn't actually "informal" (foolish, childish, devilish, for example, are well-established).

According to OED, the usage extended from nouns such as fool, child, devil centuries ago, to include adjectives - initially with colours (bluish, greenish, etc.), but "in later use also with other adjectives, and now, in colloquial use, possible with nearly all monosyllabic adjectives, and some others".

OED's definition for the suffix when applied to a noun says...

Of or belonging to a person or thing; of the nature or character of.
...
In recent colloquial and journalistic use, -ish has become the favourite ending for forming adjs. for the nonce (esp. of a slighting or depreciatory nature) on proper names of persons, places, or things, and even on phrases, e.g. Disraelitish, Heine-ish, Mark Twainish, Micawberish, Miss Martineauish, Queen Annish, Spectator-ish, Tupperish, West Endish; all-over-ish, at-homeish, devil-may-care-ish, how-d'ye-doish, jolly-good-fellowish, merry-go-roundish, out-of-townish, and the like.

Obviously some of the more "unusual" versions could well be called "informal", and because there's often the possibility of a more "scholarly" alternative (Micawberesque rather than Micawberish, for example), some people may feel it's inherently a bit informal. But I don't really think so - it's just that the suffix is so "productive" people naturally make use of it to informally create nonce words as and when they want.

As OP seems to have discovered, rather is often used conjunction with -ish forms (including "rather childish" as well as "rather Barbieish"). Usually, it will have the sense of to some extent, or in some way, but in some contexts it might have more of the "comparative" meaning to a greater extent...

"I wouldn't say she's 'Barbieish' - rather 'Dolly Partonish', to my mind."

EDIT: Because it's become increasingly common in recent years, I'll just make special mention of -ish as used to impart a certain "vagueness" to times...

"Drop by my office tomorrow afternoon. Three-ish would be fine."
"I've nearly finished coding the new app. It should be ready for testing Wednesday-ish"

In fact, this usage has become so common, the suffix can sometimes be used on its own...

"Hi. I'm just calling to make sure you're coming early to help prepare for my party tonight."
"Stop worrying! I told you before I'll definitely be there"
"Great! See you at 6 then!"
"Ish." [hangs up]

(where that final word effectively means "I'll arrive about 6, but probably later than you're expecting.")