It has nothing to do with the plural. The main word is Fleisch and the other is Schwein or Rind. Together it's Schwein-e-fleisch (the 'e' is an epenthesis). In Rindfleisch it works without an epenthesis.
There are also other epenthesises: Hund-e-leine (dog leash), Ansicht-s-karte (picture postcard), Freund-es-kreis (circle of friends).
These epenthesises (in german Fugenlaute) are not always regular, but for example for main words with the endings -keit, -heit, -ion, -ung the epenthesis is mostly 's' : Krankheit-s-anzeichen (symptom of disease)
If you add to the main word a verb, the epenthesis is always 'e': Les-e-buch (reading book)
In substantives it's mostly (e)s or (e)n, rarely er or e.
These epenthesises were already developed in Old High German.