@m_fedorov According to your first screenshot, you have been using Built-In Render Pipeline shader with URP. Please check out the URP outline shader example scene Outline Shaders URP
.
See the FAQ under "The Outline shader displays only outlines in my URP project. Why?"
m_fedorov But it's very strange that the shaders intended for URP still don't work with the WebGL build.
You are not using shaders intended for URP and you're viewing the non-URP outline shaders example scene Outline Shaders
of the core spine-unity package instead of Outline Shaders URP
of the Spine URP Shaders package.