The font you selected has no bold, italic, math or Hungarian double-acute characters. However, it is possible to fake them.
This example is a little more verbose than it needs to be, in order to suppress all warning messages.
\documentclass[11pt,a4paper]{article}
\usepackage{newunicodechar}
\usepackage{unicode-math}
\usepackage{babel}
\babelprovide[import, main]{hungarian}
\defaultfontfeatures{Scale = MatchLowercase}
\setmainfont{RymanEco}[
Scale = 1.0 ,
Ligatures = {Common, TeX} ,
Numbers = OldStyle ,
UprightFont = *-Regular ,
BoldFont = *-Regular , % Faking it.
BoldFeatures = {FakeBold = 1.25} ,
ItalicFont = *-Regular , % Faking it.
ItalicFeatures = {FakeSlant = .15} ,
BoldItalicFont = *-Regular , % Faking it.
BoldItalicFeatures = { FakeBold = 1.25, FakeSlant = .15 } ,
Extension = .otf ]
\setmathfont{STIX Two Math}
\setmathfont{RymanEco-Regular.otf}[
range={"00-"FF, up/{Latin, latin, Digits}},
Script = Default ,
script-features={},
sscript-features={} ]
\setmathfont{RymanEco-Regular.otf}[
range=it/{Latin, latin},
Script = Default,
script-features={},
sscript-features={},
FakeSlant = .15 ]
\setmathfont{RymanEco-Regular.otf}[
range=bfup/{Latin, latin},
Script = Default,
script-features={},
sscript-features={},
FakeBold = 1.25 ]
\setmathfont{RymanEco-Regular.otf}[
range=bfit/{Latin, latin},
Script = Default,
script-features={},
sscript-features={},
FakeSlant = .15 ,
FakeBold = 1.25 ]
\renewcommand\H[1]{{\accent"02DD #1}}
\newunicodechar{Ő}{\H{O}}
\newunicodechar{ő}{\H{o}}
\newunicodechar{Ű}{\H{U}}
\newunicodechar{ű}{\H{u}}
\begin{document}
This is normal text---it's okay. Ez a normális szöveg, és ez rendben van.
\textit{This text may appear italic, but it doesn't do it. Ennek a szövegnek dőlt betűsnek kellene lennie, de nem lesz az.}
\textbf{This text must be bold. Most vastagnak kellene lennie. \textit{Bold Slanted.}}
And some math text. Végül némi matematikai szöveg. $\sqrt{2} \notin \mathbb{Q}$
\begin{equation}
e^{i\pi}-1=0
\end{equation}
\end{document}

I copied your MWE with a few minor changes, such as \notin for \nin.
This fakes italic (more pedantically, slanted) with the FakeSlant command of fontspec, and bold with FakeBold. Note that using bold will defeat the purpose of saving ink! But you can reduce the factor I gave and feel virtuous. (ETA: If so, you might add a bit of FakeStretch to make the emphasized text stand out more without using so much more ink.)
The Ryman Eco font does not have precomposed ő, or the combining accent ◌̋ (U+030B), but it does have ˝ (U+02DD). You can use that glyph as a combining accent with the \accent command. The above would not quite work with source code that uses combining accents, only with the four precomposed characters it sets up.
If you don’t like the fake italics, you can call \usepackage[math-style=upright]{unicode-math} to use upright letters in math mode. In ISO style, constants such as i, e and π should be upright anyway!
Text that mixes English and other languages this way also really ought to use language tags, although you were keeping this simple as a MWE.