Intuition and analogue of Wraith axiom from synthetic differential geometry

An analogue for smooth manifolds $M$ is: For all continuous bilinear functions $\tau: \mathbb{R}\times\mathbb{R} \rightarrow TM_p$ which are constant on the axes $\tau(r,0)=\tau(0,r)=\tau(0,0)$ for all $r \in \mathbb{R}$, there's a unique factorization through the multiplication map, i.e. there's a unique function $t: \mathbb{R}\rightarrow TM_p$ such that $t(r_1 \cdot r_2) =\tau(r_1, r_2)$.

So the geometric intuition is that all functions are continuous, and are linear at infinitesimal scales.

Axiom W is about the behaviour of the second tangent bundle - it ensures that the vertical bundle of the tangent bundle, $V(M) \subseteq T\circ T(M)$, where $V(M) = T(p)^{-1}(0)$, decomposes as the pullback of the projection $p_M: T(M) \to M$ along itself. The map $[\bullet, M]:[D,M] \to [D \times D, M]$ would be written fiberwise as $\ell(v) = \frac{d}{dt}_{t = 0} (vt)$. If you look at Robin Cockett and Geoff Cruttwell's first paper on tangent categories, you can see they spend a fair bit of time talking about the universality of the vertical lift and its relationship with the Lie bracket.

You may also want to look at the definition of the core of a double vector bundle - axiom W can also be read as saying the core of the second tangent bundle of M is $TM$.

Edit: If you look at the answer here, you can see how the lift from the fibred product $TM \times_M TM \to T^2(M)$ is written. Using infinitesimals, you would write $\gamma,\beta:D\to M, \gamma(0)=\beta(0)$ is sent to the map $D\times D \to M$ which is given by $d_1,d_2 \mapsto \gamma(d_1) + \beta(d_1d_2)$. The condition that this be the kernel of $T(p)$ is provable from property W (and a good exercise).

The square root business you mentioned looks to be how you would show property $W$ holds in the category of smooth manifolds (when rewritten as the universality of the vertical lift from the tangent category axioms).