I am an Australian citizen who spent the last few months studying in Germany. My residence permit for Germany is due to expire on 15th October 2018, but I recently obtained another visa for an internship in Portugal (Visto de Estada Temporária), which started on 1st September 2018 and is due to expire 31st August 2019. Would I need to exit and re-enter the Schengen Area in order to be recorded as being in the Schengen Area under this new visa?
Generally not. The Schengen area does not (yet) have a central database of entries and exits. Furthermore, Schengen law does not have a concept of "status" as many countries do. Rather, you have some document or combination of documents that authorizes you to remain in the Schengen area.
Type D visas (long-stay visas) and residence permits are essentially identical under Schengen law. They both allow you to stay in the issuing country as long as the document is valid, and to stay in other Schengen countries up to 90 days in any 180-day period (which you can do anyway without a visa since you hold an Australian passport). If you happen to have two such documents issued by different countries that overlap in validity, that just means that you can be in either country without using up some of your 90 days. Even so, the 90/180 rule is not systematically enforced for those who have a D visa or a residence permit.
In short, you can stay in the Schengen area until 31 August 2019, and possibly even 90 days after that. If you want to leave and reenter at any time, you can.