I found the following example with SageMath. Below is the Hasse diagram and verification in SageMath that the poset is a lattice which is not Eulerian (in fact not graded) that satisfies the desired Möbius function condition. I have nothing more enlightening to say currently, but I'll post this example since it answers the latter question about a non-Eulerian example and will perhaps helps toward an answer to the first question about characterizing such lattices.

sage: P = Poset(([0,1,2,3,4,5,6,7,8],[[0, 1], [0, 2], [0, 5], [1, 4], [1, 6], [2, 3], [2, 7], [3, 4], [4, 8], [5, 6], [5, 7], [6, 8], [7, 8]]))
sage: b = P.bottom()
sage: t = P.top()
sage: P.is_lattice()
True
sage: P.is_graded()
False
sage: all(P.moebius_function(b,x) == P.moebius_function(b,t)*P.moebius_function(x,t) for x in P)
True

I found the following example with SageMath. Below is the Hasse diagram and verification in SageMath that the poset is a lattice which is not Eulerian (in fact not graded) that satisfies the desired Möbius function condition. I have nothing more enlightening to say currently, but I'll post this example since it answers the second question about a non-Eulerian example and will perhaps helps toward an answer to the first question about characterizing such lattices.

sage: P = Poset(([0,1,2,3,4,5,6,7,8],[[0, 1], [0, 2], [0, 5], [1, 4], [1, 6], [2, 3], [2, 7], [3, 4], [4, 8], [5, 6], [5, 7], [6, 8], [7, 8]]))
sage: b = P.bottom()
sage: t = P.top()
sage: P.is_lattice()
True
sage: P.is_graded()
False
sage: all(P.moebius_function(b,x) == P.moebius_function(b,t)*P.moebius_function(x,t) for x in P)
True