There is a result in functional analysis whose first known proof uses non-standard techniques:
Theorem If a bounded linear operator $ T $ on a Hilbert space $ \mathcal{H} $ is polynomially compact, i.e., $ P(T) $ is compact for some non-zero polynomial $ P $, then $ T $ has an invariant subspace. This means that there is a non-trivial proper subspace $ W $ of $ \mathcal{H} $ such that $ p(T)[W] \subseteq W $.
The proof was given by Allen Bernstein and Abraham Robinson. Their result is significant because it is related to the so-called Invariant-Subspace Conjecture, an important unsolved problem in functional analysis. Paul Halmos, a staunch critic of non-standard analysis, supplied a standard proof of the result almost immediately after reading the pre-print of the Bernstein-Robinson paper. In fact, both proofs were published in the same issue of the Pacific Journal of Mathematics!
I believe, you are looking in a wrong place. Geometry and Topology are related, but different fields of mathematics. Same with Analysis, which you are trying to put under the same umbrella. You might eventually find a definition which is broad enough to cover all three areas, but then it will cover so much in mathematics that it becomes useless.
Here are some notions of geometric structures (on smooth manifolds) that people working in geometry and topology actually use and quite successfully. This list will not answer your question, but, hopefully, will be useful (to somebody).
Geometric structure (in the sense of Cartan, I think). If I remember correctly, these are discussed in detail in the book of Kobayashi and Nomizu "Foundations of Differential Geometry". Let $M$ be a smooth manifold. Then the geometric structure on $M$ is a reduction of the structure group of the frame bundle of $M$ from $G=GL(n, {\mathbb R})$ to a certain subgroup $H<G$. For instance, a Riemannian metric is a reduction to the orthogonal subgroup. An almost complex structure is a reduction to the subgroup $Gl(n,C)$.
Geometric structure in the sense of Ehresmann (see here), or an $(X,G)$-structure. Let $X$ be an $n$-dimensional manifold and $G$ a group (or pseudogroup) of transformations of $X$. One usually assumes that $G$ acts transitively and real-analytically, but let's ignore this. Then an $(X,G)$-structure on an $n$-dimensional manifold $M$ is an atlas on $M$ with values in $X$ and transition maps equal to restrictions of elements of $G$. For instance, complex structure, symplectic structure, flat affine structure, hyperbolic structure etc, appear this way. This notion was successfully extended to cover spaces which are not manifolds, where one relaxes the assumption that charts are defined on open subsets: These extensions appear in algebraic geometry and theory of buildings.
There is an important variation on these concepts due to Gromov, called rigid geometric structures, see:
Gromov, Michael, Rigid transformations groups, Géométrie différentielle, Colloq. Géom. Phys., Paris/Fr. 1986, Trav. Cours 33, 65-139 (1988). ZBL0652.53023.
Quiroga-Barranco, R.; Candel, A., Rigid and finite type geometric structures, Geom. Dedicata 106, 123-143 (2004). ZBL1081.53027.
An, Jinpeng, Rigid geometric structures, isometric actions, and algebraic quotients, Geom. Dedicata 157, 153-185 (2012). ZBL1286.57032.
and
Feres, Renato, Rigid geometric structures and actions of semisimple Lie groups, Foulon, Patrick (ed.), Rigidity, fundamental group and dynamics. Paris: Société Mathématique de France (ISBN 2-85629-134-1/pbk). Panor. Synth. 13, 121-167 (2002). ZBL1058.53037.
Best Answer
As explained in the linked answers, the notion of a "field with one element" is a catch all term for a system of linked ideas and phenomena throughout algebra, which aren't (yet) described satisfactorily within our current axiomatic system.
So the field with one element is not in any sense a field, or even a set with extra structure, and any proposed definition solely along these lines misses the point, which is to find an algebraic framework that explains the (observed) phenomena of interest. For instance, weakening the field axioms may allow one to build a field with one element, but it doesn't solve the problem of explaining whats actually going on. See this question Why isn't the zero ring the field with one element?
As an example, the Weil conjectures for curves state that for a smooth algebraic curve $C$ over a finite field $\mathbb{F}_q$, the number of points of $C$ defined over $\mathbb{F}_{q^n}$ differs from $q+1$ by at most $2g\sqrt{q^n}$. This is an arithmetic problem, counting the number of solutions to equations defined over finite fields, but it has a beautiful solution using algebraic geometry, utilising the $2$ dimensional geometric object $C\times C$ to do intersection theory.
To my knowledge, one of the main driving forces behind the desire for a theory of a field with one element is to replicate this argument, viewing $\mathbb{Z}$ as an algebra over $\mathbb{F_1}$ (whatever this means), if there was a sufficiently developed theory that worked as expected, so we had an "intersection theory", then an analogous argument could be used to prove the Riemann Hypothesis, which is the analogue of the Weil conjecture for curves.
So a good theory of $\mathbb{F}_1$ would allow one to make sense of $\mathbb{Z}\otimes_{\mathbb{F}_1}\mathbb{Z}$, and would be sufficiently precise to develop intersection theory and the estimates needed to make the above argument work.
This is to say, the motivations are there, but finding the right definitions to encapsulate the properties we are after is absolutely the hard part.