See FAQ: Not a LaTeX renderer. LaTeXStrings is not responsible for doing the LaTeX math rendering — that is done by MathJax, MathTeXEngine.jl, or other renderers called by the plotting package or other environment. Generally these are not full LaTeX engines and only handle a subset of the math syntax.
L"$\left | H \right | dB$" works fine for me with Matplotlib’s LaTeX renderer, via PyPlot.jl or PythonPlot.jl. So if you do Plots.set_default_backend!(:pythonplot) then presumably it will work. What Plots.jl backend are you using?
There is supposed to be a Julia frontend for Asymptote but I was unable to install it and it did not expose much of Asymptote. I think the Plots API is well designed.