I believe that keyword highlighting functionality is not supported by Azure Search provider used in your version of Sitecore. Maybe future versions will support that. Best would be to ask Sitecore Support.
It should be doable according to small note in this article but it's not implemented.
Also Azure SDK and Azure Search Properties with it's HighlightFields, HighlightPostTag and HighlightPreTag properties and Azure Search Result Model supports Highlights property similarly to Sitecore Solr implementation that you mentioned in your question.
Unfortunately in Sitecore implementation of Azure Search, I couldn't find any Highlights property implementation and therefore I believe this is for now not implemented:
Sitecore is not using any Azure SDK to do the search (in oppose to SolrNet library used for Solr). They have built their own implementation using HTTP calls so using Azure Search REST API.
1) Custom development of a new search provider using either Azure SDK or Azure REST API. This would be time consuming
2) Creating your own SearchResultsDeserializer class where you would have also Highlighting implemented:
Problem is that you cannot just substitute OOTB class with your custom as there is no config entry for it. You would need to change lot's of other classes which are marked as internal. Tough solution to implement.
3) You have another option to contact Sitecore Support and ask them whether this is planned in roadmap or ask them to add this feature to feature request list for dev team. Therefore in this case you "just" have to wait for a next version or following one (if you don't have time constrains ;-) and Sitecore will add this functionality itself. I will give +1 to have this feature implemented.