A name for testing software migrated to a new environment against original application
As mentioned in the question, there is a subtle difference between Regression testing and the situation described here.

To paraphrase Wikipedia:


  Regression testing ... is re-running functional and non-functional tests to ensure that previously developed and tested software still performs after a change. If not, that would be called a regression.


However, a situation is described here where the functionality of the code is the same -- since no new features or functionality is added, this is technically not regression testing.

However, what does need to be asserted is the behaviour of the code. If run in the same way after the re-implementation, does the code still behave in the same way, beyond still giving the same results (which would be covered in unit testing and functional test).

In "The Practical Test Pyramid", Martin Fowler describes the various types of testing which a well-maintained application could have. Near the apex are "Acceptance Tests", i.e


  Do Your Features Work Correctly?


After a re-write, I believe Acceptance Tests would be the answer here.

Note - Acceptance tests also include Operational Acceptance Tests, described by Wikipedia as:


  Operational acceptance test (OAT) criteria (regardless if using agile, iterative or sequential development) are defined in terms of functional and non-functional requirements; covering key quality attributes of functional stability, portability and reliability.