What's my TIO uniqueness?
JavaScript (Node.js),  819  779 bytes
Returns 0 for undefined TIO uniqueness.
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Try it online! (test cases)
Try it online! (all languages)
How?
There are only \$10\$ distinct TIO uniqueness values, ranging from \$0\$ to \$13\$. Besides, more than 50% of the languages have a TIO uniqueness of \$3\$.
 uniq. | count | ratio
-------+-------+--------
   0   |   97  | 14.24%
   1   |   12  |  1.76%
   2   |  140  | 20.56%
   3   |  357  | 52.42%
   4   |   56  |  8.22%
   5   |   11  |  1.62%
   6   |    2  |  0.29%
  10   |    4  |  0.59%
  11   |    1  |  0.15%
  13   |    1  |  0.15%
We need:

a function \$H_0\$ that turns the language name \$s\$ into an integer \$H_0(s)\$
a function \$H_1\$ and a corresponding lookup table \$T\$ so that the TIO uniqueness of the language is \$T[H_1(H_0(s))]\$

It seems unlikely to find some simple functions \$H_0\$ and \$H_1\$ that would directly lead to a small lookup table. But if we use \$3\$ (the most common uniqueness) as the padding value for unused slots, the table is going to contain a lot of \$3\$'s.
So our strategy is to use:

the first 14 printable ASCII characters (32 to 45, i.e.   to -) to encode the actual uniqueness values other than \$3\$
all other printable characters (46 to 126, i.e. . to ~) to encode runs of 2 to 82 consecutive \$3\$'s

Conveniently, the single quote ' (ASCII code 39) will never appear in the table because no language has a uniqueness of 7. So we can use it as our string delimiter and safely use &quot; and ` in the table without having to escape them.
We now need to find \$H_0\$ and \$H_1\$ so that the distribution of actual uniqueness values among the \$3\$'s leads to the smallest table with this simple compression scheme.
After some brute-forcing, it turns out that the following functions work reasonably well:

Buffer(s).reduce((p, c) =&gt; p * 79 ^ c) &gt;&gt;&gt; 0 (actually implemented with a .map())
% 12635