The word "uniquer" is technically not incorrect, but it is not commonly used in standard English.

The word "unique" is already a superlative, meaning it describes something that is one of a kind or singular, having no equal or equivalent. As such, it does not require the addition of "er" to make it comparative (e.g., "uniquer") as you would with regular adjectives like "taller" or "faster."

The use of "more unique" is generally discouraged in standard English because "unique" itself already denotes something as being one-of-a-kind and unparalleled. Adding "more" to "unique" can be seen as redundant or logically awkward, as there are no degrees of uniqueness—it is an absolute quality.

In colloquial or informal speech, you might occasionally hear people use "more unique" without strict adherence to grammar rules. However, in formal writing and speech, it is better to avoid using "more unique."

Instead, when making comparisons with "unique," it is more appropriate to rephrase the sentence to express the distinction differently. For example:

  1. Of all the designs, this one stands out as the most distinctive.
  2. Among the three rare species, the first one is the most exceptional.
  3. Her approach to problem-solving is more original than her peers' methods.

Example sentences

  1. The ancient artifact displayed in the museum is truly unique, as no other piece like it has ever been discovered.
  2. Her style of painting is so unique that it instantly captivates anyone who sees her work.
  3. The small coastal town has a unique charm, drawing tourists from all over the world.
  4. Each snowflake that falls from the sky is unique, with its own intricate pattern.
  5. The boutique offers a collection of unique handcrafted jewelry, perfect for those seeking something special.
  6. The restaurant's menu features a unique fusion of flavors from different cuisines, making it a delightful culinary experience.
  7. Growing up in a multicultural household gave her a unique perspective on life and traditions from different backgrounds.
  8. The endangered species of bird found in this region has a unique call that distinguishes it from other birds in the area.
  9. The company's marketing strategy was unique and innovative, setting it apart from its competitors.
  10. The old bookstore has a unique atmosphere, filled with the scent of aged books and the warmth of knowledge.

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Reference:

unique
1. being the only existing one of its type or, more generally, unusual, or…
Cambridge Dictionary
Are “more unique” and “less unique” acceptable phrases?
Answer (1 of 10): The problem with these phrases is that they imply that there there are degrees of uniqueness, whereas technically uniqueness is a quality that is either present or absent (vs being present to some degree). Equivalent words / phrases that avoid this problem include: “rarer” (in…
Quora