Manual says, that Tangent node is to be used with Anisotropic BSDF https://docs.blender.org/manual/en/dev/render/shader_nodes/input/tangent.html

@Jeroen Bakker (jbakker) is this limitation of Tangent node?

Will need to investigate. What I miss in the report whether there is a difference between eevee/cycles.
If cycles doesn't support it it is mostly by design, if it is an error in eevee it could be a bug, known issue or limitation.

Checked with cycles, and it doesn't work there. Node produces output that looks usable when plugged into color socket though.

While the bump node does produce output that looks usable it's actually nothing more than the geometry normals i.e the bump node is not modifying the geometry normals in any way and just returning them back as is. You can check this by doing a vector math 'distance' operation with geometry normals followed by a math compare node, if you compare it with 0 even with 0 epsilon both evee and cycles will tell you that they are equal though at very high strength values you get an interesting looking output, i am guessing it's just floating point inaccuracies manifesting at very high values, the weird thing though is if it's not modifying the normals in any way then why are floating point inaccuracies all of a sudden manifesting??

This is essentially the same as T86375: bump node ignores height input (with certain inouts used as mapping vectors in the tree, seems to require `SOCKET_OUT_POINT`as opposed to `SOCKET_OUT_NORMAL`), will merge these reports