Using TaskLocalRNG results in identical noise across processes

When multiple noise processes are created sequentially with the default TaskLocalRNG or a user passed TaskLocalRNG and no intervening rand calls, copy(rng) gives each process an identical RNG state, resulting in statistically identical samples.

using DiffEqBase, DiffEqNoiseProcess

# Create two WienerProcesses back-to-back with default RNG
W1 = WienerProcess(0.0, 0.0, 0.0; reseed = false)
W2 = WienerProcess(0.0, 0.0, 0.0; reseed = false)
sol1 = solve(NoiseProblem(W1, (0.0, 1.0)); dt = 0.1)
sol2 = solve(NoiseProblem(W2, (0.0, 1.0)); dt = 0.1)

println("W1 samples: ", sol1.W)
println("W2 samples: ", sol2.W)
println("Identical?  ", sol1.W == sol2.W)

Output (v5.27.0): Identical? true — both processes produce the exact same noise samples.

This is due to the changes made in https://github.com/SciML/DiffEqNoiseProcess.jl/commit/c305243aa484cd4e7946e6ccf301852bc6e418b7

Start with the WienerProcess constructors and the TaskLocalRNG path described in the reproduction, then trace how copy(rng) is used when processes are created back-to-back. Re-run the provided NoiseProblem example and confirm that sequential processes no longer produce identical samples.