The primary questions raised are:
Is the "translation" correct? If the first sentence is right (it
probably is), are there other examples of such "weird" sentences?
Is it the same as in the shampoo bottle example? That the bottle is
way better than the other bottles, so they don't need to bother
"applying" to a "shampoo bottle competition" (or similar), because
they will lose?
This last question is reasonably close to the actual intended meaning (as inferred from the entire shampoo bottle label context).
An important unstated premise is understanding that marketers use every word, phrase, typeface, color, etc. on labels in an attempt to improve the chances of success in the marketplace. The sentence in question is of a persuasive nature designed to increase sales by targeting the consumers both in the store (prospective purchaser) and in the shower (previous and hopefully loyal future purchaser).
Many of these answers focused on the origin of the phraseology, a question that wasn't asked and only indirectly informs an answer to the questions asked.
The statement Ordinary shampoos need not apply! communicates a number 1st level messages:
- This shampoo is not ordinary
- (To use the questioner's wording) This shampoo is, in fact, so extraordinary that "ordinary" shampoos wouldn't even qualify to compete in a shampoo contest in which this shampoo was competing.
- Given the phrases common usage in job postings, however it's something akin to "Announcement regarding the open position for the role of washing this consumer's hair. The shampoo in hand is so far superior to the other potential applicants that it would be a wasted effort for them to apply and therefore of the consumer's time to consider or 'interview' them."
- An alternate reading is as a taunt from one potential applicant to all other shampoos.
As other answers have indicated, there is an additional layer of meaning with the other relevant meanings of the word "apply" describing the process of using shampoo on one's hair. Thus the alternate sub-meaning is that this shampoo negates the need for other shampoos to be "applied" to one's hair.
It finally does all of the above while, arguably, subtly raising the question to consumer standing the store aisle reading a shampoo label whether their time is being well spent on this task in light of the fact that the shampoo in hand is so far superior to all others.
To go further down the advertising-lens analysis rabbit hole, the tone is intended to form an emotional bond with the consumer by asking them to join this shampoo in subtly mocking the pretentious or just overly complex wording of other shampoos' labels. This one will "shoot straight" and talk plainly, so therefore should be trusted.
The consumer is thus smarter by saving time in not evaluating other inferior products AND feels good about joining forces with "cool" shampoo on the aisle that is not afraid of calling out the other uncool shampoos.