[C++][Flight] Remove the need for serialization_internal.h inside python/flight.cc

[asfimport]

Making a separate JIRA issue for the work done in #13311 where the need for serialization_internal.h inside python/flight.cc was removed.

See:

#13311 (comment)

https://github.com/apache/arrow/pull/13311/files#diff-b4f0e7ff0c555ddec5cb880a8b636cdcce25453ec3fb998d0026f2b8bd5625f7

https://github.com/apache/arrow/pull/13311/files#diff-8b9efe1f8818751a9c88f12fe6c81138bfa673f965bbf8ee781353fc834fd49c

https://github.com/apache/arrow/pull/13311/files#diff-4b08b8c4f93b3a02a872894b39e2d35e60af11669f4dcad6c212f85a41343e95 

Reporter: Alenka Frim / @AlenkaF
Assignee: Alenka Frim / @AlenkaF

Related issues:

• [C++][Flight] Remove the need for serialization_internal.h inside python/flight.cc (duplicates)

_{Note: This issue was originally created as ARROW-17012. Please see the migration documentation for further details.}