What you call a *Fano plane on $\kappa$* is usually simply called a *projective plane* of cardinality $\kappa$ (with set of points equal to $\kappa$). Pooter's answer uses the fact that any field of size $\kappa$ gives rise to a projective plane of size $\kappa$, which is the easiest way to answer the question. I want to give a more general model-theoretic perspective on the question as well as your follow-up question about uniqueness in the comments.

The projective plane axioms can be written down as a first-order theory $T$, and $T$ has at least one infinite model, so by the Löwenheim-Skolem theorem, it has a model of every infinite cardinality $\kappa$. Moreover, if we let $T_\infty$ be $T$ together with axioms saying the domain is infinite, then $T_\infty$ is not a complete theory (for example, the Desargues axiom and its negation are both consistent with $T_\infty$). So $T_\infty$ has multiple non-isomorphic (even non-elementarily equivalent) models of every infinite cardinality.

To say that a first-order theory $T$ has a unique model up to isomorphism in some uncountable cardinality $\kappa$ is to say that $T$ is *uncountably categorical*. In a sense, Morley's theorem says that this property is very unusual. It certainly doesn't hold for the theory of projective planes. But it does hold for a completion of this theory, namely the theory of projective planes over algebraically closed fields of characteristic $0$ (indeed, projective planes over fields are bi-interpretable with their base fields, and the theory of algebraically closed fields of characteristic $0$ is uncountably categorical - the same is true for fixed characteristic $p$).

In this recent paper, Gabe Conant and I isolate a particularly model-theoretically interesting class of infinite non-Desarguesian projective planes (the *generic* planes), which are the models of a complete first-order theory $T_{2,2}$. Among other things, we show that there are continuum-many countable generic planes up to isomorphism.