Let $t : a \to a$ be a monad in a weak 2-category. According to nLab, the 1-dimensional universal property of a Kleisli object $(f_t : a \to a_t, \lambda : f_t t \to f_t)$ is that for any right $t$-module on $x$, $(r : a \to x, \alpha : r t \to r)$, there is a unique morphism $g : a_t\to x$ such that $r = g f_t$ and $\alpha = g \cdot \lambda$.
But, in the case of a weak category, for 1-cells we do not want equalities (as it would be "evil") but we want isomorphisms instead. This means that the equality $r = g f_t$ should be replaced by an isomorphism between $r$ and $g f_t$. Now, shouldn't this isomorphism be unique? Could this be proved using the 2-dimensional universal property of a Kleisli object? Or do we need to add a coherence axiom to the definition?
P.S. See this answer for a more detailed definition of a Kleisli object than the one in nLab.
Although the definition of Kleisli object may appear "too strict", this is not actually the case. Kleisli objects are flexible colimits, which means that they are also bicolimits, and hence also have the pseudo universal property you describe. To answer your question, the invertible 2-cell is part of the data of the bicolimit, and is not generally unique. However, it is required to satisfy the usual coherence property inherent in the definition of a 2-dimensional colimit.