Customizing Logo on "Site" and "Sub-sites"

I'm trying to better understand your requirement here.

You want a custom logo on both site and subsite but you don't want logo to inherit from parent site correct?

Please make sure address field is clean and empty in site title logo for susbite to make sure it doesn't inherit parent logo.

You can add below alternate css to your subsite and set your site logo to image you would like.

img[src*='siteIcon'] {
display: block;
-moz-box-sizing: border-box;
box-sizing: border-box;
background: url(‘/Style Library/mySiteLogo.png’) no-repeat; /* Path to the site logo */
width: 180px; /* Width of new image */
height: 64px; /* Height of new image */
padding-left: 180px; /* Equal to width of new image */
}

.ms-siteicon-img, .ms-siteicon-a {
max-height: none;
max-width: none;
}