Solution for individual objects

(Solution for mesh islands at the end of the answer)

Use this to copy material to all objects: Copy material to another object? (where the "special" in newer blenders is a "down arrow")

(the same principle for copying apply to geometry nodes, in blender 4.2, you click the down arrow on the material or the modifier)

Geometry nodes

These store a X axis for the texture.

• It needs tiles with 4 edges
• It takes the two smallest edges and calculate X based on the center of these edges
• It stores the normalized X axis in a named attribute (important: store as vector)

Shader nodes

• Take the Texture X axis
• Calculates the Texture Y axis by cross product Normal x X
• Calculates the position of the geometry in the new texture coordinate system (dot products)

(The wave texture represents your wood texture)

Result

Solution for mesh islands

Here, you have a single object instead of many individual objects.

• This requires that all islands have 4 vertices!!