You can't have both at the same time, and Rancher only applies to animal products, not artisan goods from animal products. (If you pick Rancher at level 5, you will get the choice to pick between coop master and shepherd at level 10.
Does rancher and artisan stack multiplayer?
Stardew Valley Rancher Or Tiller Multiplayer
Skill bonuses and professions do not stack. Stardew Valley will choose the highest bonus when selling items. ... Never forget to enjoy, whether single-player or multi-player game mode.
Do professions stack in Stardew Valley?
Skill bonuses and professions do not stack, and the game will pick out the highest bonus when selling items. This means it is wide to spread out your specializations for maximum profit, for example, have at least one person pick “Rancher” while the other picks the “Tiller” profession.