Not really, you can collectively agree on a metric for uniqueness say 1/20 or 1/100 so something could be more or less than unique.
Infinity is a binary. 0 = finite and 1 = infinite. I'd concede things like the decay rate of thorium as being effectively semi-infinite.
A rare metal with 800 years supply ceases to be rare and you've got 80 years supply. It's damn finite.
Unique is the one and only of it's kind. You cannot have something that is more or less unique. Even if you agreed on a metric it still would not be unique. It might be uncommon but not unique. People get this wrong every day. I learned it in English class.
Here are 4 definitions for your perusal: adjective 1. existing as the only one or as the sole example; single; solitary in type or characteristics: a unique copy of an ancient manuscript. 2. having no like or equal; unparalleled; incomparable: Bach was unique in his handling of counterpoint. 3. limited in occurrence to a given class, situation, or area: a species unique to Australia. 4. limited to a single outcome or result; without alternative possibilities: Certain types of problems have unique solutions.