Stage History are available in both classic and lightning if they are added to the page layout.
In your case, it is not coming to Lightning. The possible reason is:-
There are two component available in Lightning app builder that can show related list on a lightning page. They are:-
- Related List
- Related List-Single
Realted list will show all the related list available on the page together where as Related list single allows to select single related list and show it alone.
You might be using Related list single component. You can add one more related list single component with showing stage history.