As you said, url-loader does not handle urls in style files. To do that, you need to use postcss-url package.
To use a custom webpack configuration without ejecting, you need to install this package. After configuring your workspace with the directions provided in that package, you can use the following webpack configuration:
const url = require('postcss-url');    
const urlPlugin = url([{ filter: '**/*.svg', url: 'inline' }]);
    
module.exports = config => {
  for (const rule of config.module.rules) {
      const loader = rule.use && rule.use.find(x => x.loader === 'postcss-loader');
      if (loader) {
          loader.options.plugins = [urlPlugin];
      }
  }
    
  return config;
};
Basically, this code adds postcss-url plugin to every postcss-loader in Angular's default webpack configuration.
You can edit this configuration to customize for your needs. For example, you can use maxSize parameter to exclude files greater than a certain size. Read postcss-url repo for all options.
EDIT by Zygimantas:
I am accepting your answer as correct and adding a Typescript version of webpack.config.ts
import { Configuration, NewLoader } from 'webpack'
import * as PostCssUrlPlugin from 'postcss-url'
export default (config: Configuration) => {
    if (config.module === undefined) {
        throw new Error()
    }
    let patched = false
    for (const rule of config.module.rules) {
        if ('use' in rule && Array.isArray(rule.use)) {
            const loader = rule.use.find(
                (a): a is NewLoader =>
                    typeof a !== 'string' &&
                    'options' in a &&
                    a.loader !== undefined &&
                    a.loader.includes('postcss-loader'),
            )
            if (loader !== undefined && loader.options !== undefined) {
                loader.options.plugins = [PostCssUrlPlugin([{
                  filter: '**/*.svg',
                  url: 'inline',
                }])]
                patched = true
            }
        }
    }
    if (!patched) {
        throw new Error('Could not patch webpack configuration')
    }
    return config
}
NOTE:
In Javascript version, I had to use x.loader.includes('postcss-loader') instead of x.loader === 'postcss-loader' because loader value was a full path in my case.