html
css
.logo-img path {
You could set your SVG as a mask. That way setting a background-color would act as your fill color.
HTML
CSS
.logo {
background-color: red;
-webkit-mask: url(logo.svg) no-repeat center;
mask: url(logo.svg) no-repeat center;
}
JSFiddle: https://jsfiddle.net/KuhlTime/2j8exgcb/
MDN: https://developer.mozilla.org/en-US/docs/Web/CSS/mask
Please check whether your browser supports this feature: https://caniuse.com/#search=mask