Filtering objects according to required and preferred attributes
The most important piece of advice I can offer is get rid of any. By using any you are making Typescript ignore potential errors in your program. opportunity could be described with the following interfaces:

interface Opportunity {
  id: number
  title: string
  nationalities: Criteria[]
  skills: Criteria[]
  backgrounds: Criteria[]
  languages: Criteria[]
}

interface Criteria {
  option: 'preferred' | 'required'
  id: number
  name: string
}


Furthermore, the backgrounds, skills, languages, and nationalities properties should be correctly typed. In this case, I'd just add another interface, SeparatedCriteria isn't really the best name, but you can probably come up with a better one as you are more familiar with what each piece of data represents. 

interface SeparatedCriteria {
  preferred: Criteria[]
  required: Criteria[]
}


filterByProperty has a lot of redundancy. At the very least, I would recommend defining isRequired and isPreferred local functions to help cut down on some of the noise.

However, this is still rather noisy especially as the logic is the same for each property - the only thing that changes is the key. With this in mind, we can define an array of keys, ['skills', 'backgrounds', 'languages', 'nationalities'] and assign this[key].preferred and this[key].required.

The trick to making this work correctly is to use both intersection types and index types. It is possible to do this without knowledge of these topics, but the solution will not be type safe as it would be possible to add keys which aren't present in both Opportunity and AppComponent without receiving a compiler error. Here's how I would implement this function. You could now get rid of isRequired and isPreferred as they are only used once, but I believe it makes the logic more clear to leave them as is. 

filterByProperty() {
  const isPreferred = ({ option }: Criteria) =&gt; option == 'preferred';
  const isRequired = ({ option }: Criteria) =&gt; option == 'required';

  const propertyKeys: Array&lt;keyof AppComponent &amp; keyof Opportunity&gt;
    = ['skills', 'backgrounds', 'languages', 'nationalities'];

  for (const key of propertyKeys) {
    this[key].preferred = this.opportunity[key].filter(isPreferred);
    this[key].required = this.opportunity[key].filter(isRequired);
  }
}