This is due to how the depth offset factor is defined, it will not have any effect on triangles which are aligned with the camera plane.

The depth offset factor scales with respect to the Z slope of the polygon. When using the 'View' particle system Renderer Alignment, the particles are aligned with the camera plane. This means all of the particles will have a Z slope of 0. Thus, the effect of the depth offset factor will be eliminated, as it will be multiplied with 0.

In the other particle system render alignments, the particle polygons are not perfectly aligned with the camera plane and so there is a z-slope and so the depth offset factor will have an effect with these.

See https://docs.unity3d.com/Manual/SL-CullAndDepth.html
"Factor scales the maximum Z slope, with respect to X or Y of the polygon"

For detailed reference of exactly how this works in GPU hardware, see the graphics API specifications:

In OpenGL, the depth bias Factor and Units correspond to the glPolygonOffset(Factor, Units). See section 14.6.5 of the OpenGL 4.6 core reference.

In D3D11, the depth offset Factor corresponds to the SlopeScaledDepthBias field of D3D11_RASTERIZER_DESC. See
https://docs.microsoft.com/en-us/windows/win32/direct3d11/d3d10-graphics-programming-guide-output-merger-stage-depth-bias
or https://microsoft.github.io/DirectX-Specs/d3d/archive/D3D11_3_FunctionalSpec.htm#15.10%20Depth%20Bias
for a reference on how this functionality works.

For Vulkan, see 24.7.3 - Depth Bias https://www.khronos.org/registry/vulkan/specs/1.2/html/chap25.html#primsrast-depthbias where the depth offset Factor corresponds to depthBiasSlopeFactor.