I figured out how you can achieve it. I used TextMate syntax format added to tokenColorCustomizations configuration within VS Code's settings.json. So this setting:
"editor.tokenColorCustomizations": {
"textMateRules": [
{
"name": "CSS Class and Support - Property Name",
"scope": [
"source.css support.type.property-name",
"source.sass support.type.property-name",
"source.scss support.type.property-name",
"source.less support.type.property-name",
"source.stylus support.type.property-name",
"source.postcss support.type.property-name"
],
"settings": {
"foreground": "#9CDCFE"
}
},
{
"name": "CSS Class and Support - Property Value",
"scope": [
"source.css support.type.property-value",
"source.sass support.type.property-value",
"source.scss support.type.property-value",
"source.less support.type.property-value",
"source.stylus support.type.property-value",
"source.postcss support.type.property-value"
],
"settings": {
"foreground": "#CE9178"
}
},
{
"name": "CSS Class and Support - Class",
"scope": [
"source.css entity.other.attribute-name.class.css",
"source.sass entity.other.attribute-name.class.css",
"source.scss entity.other.attribute-name.class.css",
"source.less entity.other.attribute-name.class.css",
"source.stylus entity.other.attribute-name.class.css",
"source.postcss entity.other.attribute-name.class.css"
],
"settings": {
"foreground": "#D7BA7D"
}
}
}
would end up with that appearance inside css alike files:

For more information refer to Syntax Highlight Guide.