. I am wondering whether this "breaks" the advantages of the strategy design pattern, or whether this makes any difference?
Copying code is bad; if you have to fix it in one place, then chances are that you will forget to copy the fix into all copies. Think of the poor soul who has to support this mess!
If the strategies have common code then you can either
put common code into a utility class and add aggregate / add the common class as member of both implementations,
put common code into a base class and inherit this class in both implementations
Aggregation is usually favored over inheritance.