"Nobody seems to define the term anywhere."
The term "universal quantum algorithm" is not commonly used. In the case of factoring numbers it could mean one of two things:
Some algorithms for factoring numbers do not exploit "universal quantum computers" meaning that they could work on simpler quantum annealers like the D-Wave machine which cannot do universal quantum computation. These would not be "universal" quantum algorithms, and ones that do require universal quantum computers could (although it wouldn't be a very common way to say it) be called "universal".
Some algorithms for factoring numbers cater only to specific types of numbers (e.g. semiprime numbers, or numbers of the form $(p-1)(q-1)$ for primes $p$ and $q$). These include:
- Trial division
- Wheel factorization
- Pollard's rho algorithm
- Algebraic-group factorization algorithms
- Fermat's factorization method
- Euler's factorization method
- Special number field sieve
In the paper that you mentioned, the algorithm can run on a non-universal D-Wave machine (even though they used QAOA on a circuit-based machine to solve the QUBO part of their algorithm, instead of just running it on a D-Wave machine), they more likely to mean the second option above. They mean that their algorithm will work for the factorization of any integer.
I recently wrote a much longer explanation of that paper here, in the answer to Quantum Computing Used to Break RSA by "fixing" Schnorr's Recent Factorization Claim?.