Sets of lengths do not characterize numerical monoids.

By Jeff Amos, Scott T. Chapman, Natalie Hine, J. Paixão. 2007

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Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids (2010, unknown): “In [2], S.”
Atomicity and boundedness of monotone Puiseux monoids (2016, background): “ds inﬁnitely generated by well-behaved sequences of rationals. Numerical semigroups generated by intervals, arithmetic sequences, and generalized arithmetic sequences have been intensely studied (see [6, 2, 11] and the references therein). In particular, the simplicity of arithmetic sequences has fa”
On the delta set and the Betti elements of a BF-monoid (2012, background): “Moreover, the papers [2,5,9–11] all consider properties related to the Delta set (or (S)) of a numerical monoid S.”
Factorization invariants of Puiseux monoids generated by geometric sequences (2019, background): “The system of sets of lengths of numerical monoids has been studied in [1] and [23], while the system of sets of lengths of Puiseux monoids was first studied in [30].”
On strongly primary monoids, with a focus on Puiseux monoids (2019, background): “In special cases (such as for numerical monoids generated by arithmetic sequences or Puiseux monoids generated by geometric sequences) very explicit descriptions of sets of lengths and their unions are available (in these cases, the initial and end parts are empty; see [1, 9, 11]).”