On the realisation of monoids as syntactic monoids of languages
It seems there is a paper answering this exact question, and even in the more general case of $\omega$-regular languages, but I cannot find an open-access version. If somebody finds a link without paywall it would be great. I requested the full-text on ResearchGate.

Title: Which Finite Monoids are Syntactic Monoids of Rational omega-Languages.

Authors: Phan Trung Huy, Igor Litovsky, Do Long Van

Abstract: A notion of ω-rigid sets for a finite monoid is introduced. We prove that a finite monoid M is the Arnold's syntactic monoid of some rational ω-language (ω-syntactic for short) if and only if there exists an ω-rigid set for M. This property is shown to be decidable for the finite monoids. Relationship between the family of ω-syntactic monoids and that of ∗-syntactic monoids (i.e. the syntactic monoids of rational languages of finite words) is established.



Additionally, the wikipedia page on syntactic monoids states:


Every finite monoid is homomorphic to the syntactic monoid of some non-trivial language,[1] but not every finite monoid is isomorphic to a syntactic monoid.[2]
Every finite group is isomorphic to the syntactic monoid of some non-trivial language.[1]


[1] McNaughton, Robert; Papert, Seymour (1971). Counter-free Automata. Research Monograph 65. With an appendix by William Henneman. MIT Press. p. 48. ISBN 0-262-13076-9. Zbl 0232.94024.

[2] Lawson (2004) p.233