But is there anything physical that connects the electric field at a point to that at its nearby points (just like we had for strings) ?

Yes, the field itself is spatially connected. In Maxwell’s vacuum equations the spatial connection between nearby points is given by the expressions $\nabla \cdot$ and $\nabla \times$ in the differential form of the equations.

$\nabla \cdot$ describes how a field emanates from a location and diverges to neighboring locations. So $\nabla \cdot \vec E=0$ and $\nabla \cdot \vec B=0$ mean that neither the E field nor the B field emanate from anywhere in the vacuum.

$\nabla \times$ describes how something curls around the neighboring locations. So $\nabla \times \vec E = -\partial \vec B /\partial t$ and $\nabla \times \vec B = \mu_0 \epsilon_0 \ \partial \vec E/\partial t$ mean that if either field changes over time at one point the other will curl around the neighboring points.

But is there anything physical that connects the electric field at a point to that at its nearby points (just like we had for strings) ?

Yes, the field itself is spatially connected. In Maxwell’s vacuum equations the spatial connection between nearby points is given by the expressions $\nabla \cdot$ and $\nabla \times$ in the differential form of the equations.

$\nabla \cdot$ describes how a field emanates from a location and diverges to neighboring locations. So $\nabla \cdot \vec E=0$ and $\nabla \cdot \vec B=0$ mean that neither the E field nor the B field emanate from anywhere in the vacuum.

$\nabla \times$ describes how something curls around the neighboring locations. So $\nabla \times \vec E = -\partial \vec B /\partial t$ and $\nabla \times \vec B = \mu_0 \epsilon_0 \ \partial \vec E/\partial t$ mean that if either field changes over time at one point the other curls around the neighboring points.