Perhaps I don't have a full appreciation as a non South Australian (or I just differ in what I think constitutes a rivalry) but I don't think Adelaide vs Port should be No 1.
Here is my rationale:
1. They have never played off for the ultimate prize - Geelong and Hawthorn as just one example have played in 4 GF's, one many consider the greatest of all time and another the biggest upset of all time. Coll and Carl have played in MANY, including biggest comeback and other storied controversies, Ess v Haw (3 straight), etc. Never once playing for a flag surely takes some heat off.
2. They have only played in 1 final ever, a boring 83 point thrashing of 1st vs 8th. Basically, there have been no games of any real consequence between the 2. Compare that to other rivalries above where there have been numerous close finals, heart breaks, etc.
3. As far as I can recall, there have been no significant spiteful clashes between the 2. Hawthorn and Essendon had the 'line in the sand' brawl and the Lloyd/Sewell fiasco after a long line of incidents, Coll and Carl have had a number of notorious brawls, including arguably the biggest ever (which was in a GF). Geel/Haw has Matthews/Bruns, 89GF (several famous incidents), etc. On field has always been pretty meek between these two as far as I can see.
4. No storied off field controversies between the 2 clubs themselves (at least not on a national level that I can recall) - Geel and Haw had Kennet's curse, Chappy's pact and a bunch of others, Ess vs Haw have had heaps including police involvement, etc. Can't even think of any big trade controversies between the 2.
5. Relatively short history - Whilst I understand that Port were a dominant force in the SANFL, Adelaide didn't even exist in that competition. Port's rivalries were with Norwood, Sturt, West Torrens, etc which are separate clubs entirely and still playing in the SANFL (incidentally against Port). Hell, I know many Port supporters who supported Adelaide in the AFL and Port in the SANFL until Port were admitted.
6. No real history of close hard fought encounters or even being good at the same time - Whilst 2018 produced 2 crackers between port and Adelaide, go back 5 or 6 games and the average margin blows out past 5 goals (including the 2 close ones from last year). Until last year, the teams had only played twice EVER in matches decided by less than a goal (in 40 something 'showdowns')! and neither have finished in the top 4 together. Compare that to say, Hawthorn and Geelong, who were both basically top 4 for a decade and had about 15 games in a row decided by 2 goals or less (something like that anyway).
The only rivalry I see is that they share a footy mad town and so supporters of both clubs enjoy the demise (or 'one up manship') of (or over) the other. Nothing really between the clubs themselves and certainly no different to basically all the other rivalries where supporters dislike each other.