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There are non-co/unital Frobenius algebras in Hilb which do not posses a unit. Consider for example, the image of a an infinite dimensional semi-Frobenius algebra in Pinj under the ell^2 functor. This has no unit, as Frobenius algebras in Hilb must be finite dimensional.
Just if you are looking for more references, I believe that the proof of this fact is a slight modification of Proposition 2.6 in one of my go to papers Some algebraic laws for spans (and their connections with multirelations), by Bruni and Gadducci which exhibits an equivalence between spans of (infinite) sets and multirelations.
