“Suck suck ice” does appear to be the Chinese meaning, but when translating a literal translation might not always work. In this case, “suck suck ice” does not sound appealing, but “want want ice” at least has a similar way of getting the product’s name across.
“Suck” in English does have the additional meaning of being a terrible product (this sucks!), so I can see that being a reason for the “translation” being changed
So, the reason it’s ‘Want Want Ice’ is that the brand is 旺旺, which is literally the character for ‘flourishing’ twice. It’s a food company from Taiwan, apparently. (I knew the brand because I used to have biscuits made by them quite often as a kid, but I never knew where they were made. You don’t really bother about these things as a child.) In Mandarin, those characters are pronounced wàng wàng (wah-ng wah-ng, both pronounced with a sharp, downward tone, if anyone’s wondering), but I guess those sounds don’t mean anything in English (and frankly, looking at the reactions to ‘suck suck’ on this thread, I can already imagine what people would be thinking seeing ‘wang wang’). Hence, well… ‘Want Want’. Customers should desire a company’s products, so… there we go. It also sounds a bit like a child whining about wanting something, which fits the company’s target audience.
As for the whole 吸吸 (xī xī – pronounced ‘see see’ in a flat tone at a pitch slightly higher than one’s usual speaking voice) business, it’s like how onomatopoeia get repeated in Japanese. It’s meant to sound a little childish and cute, and helps the brand to get the idea across while avoiding technical terms, which makes the product child-friendly and fun. (Can you imagine ‘Crushed Fruity Water Ice with Straw Attached’ as the name for this? Because that’s probably what we’d have if Chinese didn’t have these cutesy alternatives.) It’s quite common to see such repetition on children’s products, and it’s not rare to hear Chinese-speaking parents doubling words when speaking to their kids: ”把手手给我。” (literally ‘[object] hand-hand give me’ i.e. ‘give me your hand/hands’). Chinese for adults/older children would just use one 手. Of course, we can go into infantilising things in order to create a certain sort of innuendo, which is possible in any language, but that’s not relevant here because Want Want is a brand known for products popular with children like sweets and biscuits. It’s like saying「ほれほれ」to a cat when giving it food in order to mean ‘here you go’ instead of abruptly stopping at「ほれ」or「ほら」. Repeating characters is common in Chinese in order to create a certain effect, most often emphasis. Other possible meanings include plurality (人々 isn’t exclusive to Japanese: in Chinese, 人人 means ‘everyone’ or ‘most people’ within a certain set of people as well) or actual repetition of an action.
How I know all this: I’m a Chinese speaker and I grew up in Southeast Asia with brands like this one around me.
PS: Didn’t mean to be a party pooper since everyone seemed pretty amused by ‘suck suck ice’, but it’s just that it’s super innocent-sounding in Mandarin. The only context in which it would be ahem ahem would be a context in which ‘suck suck’ is obviously meant to mean… something else. But yeah, I guess that’s why the product name just couldn’t be translated normally.