This is why color management was invented. It solves exactly these problems.
Work in an appropriate color space (probably sRGB for this), make sure the icc profile is embedded in the file, and use only color managed software to view the file. That includes web browsers, image viewers and so on. Most of them are not color managed - so find those that are.
This is the only way to get color consistency. Color managed applications should always display identically.
If you exporting to PNG you have an option to embed a color profile, but not for SVG.
Neither Photoshop or Illustrator let you embed a profile with an SVG. Looks like there is some support for profiles in SVG coding, but in the end you should just color correct in sRGB either via an assignment or setting your RGB working space to sRGB.