Do the market place / fishing guy / photo guy / dog tag quests in FFXV ever end?

Yes, the quests you mentioned do have a definitive end. I will summarize the ones you alluded to below, but a comprehensive list of all side quests can be found here.

Food Retrieval (retrieved from Takka in Hammerhead): The market quests end with the A Meat Most Magnificent quest.

Fishing (retrieved from Navyth the fisherman in Duscae): The fishing side quests end with the Angler's Nightmare quest.

Photos (retrieved from Vyv in Lestallum): the photo quests end with the Cursed Canvas Quest, and a full list of these photo quests can be found here.

Dog Tags (originally retrieved from Dave in Longwythe Rest Area, though his location changes as you progress through his side quest): the dog tag quests end with Witch of the Woods. A full list of the dog tag quests, as well as Dave's location to initiate each quest, can be found here.

Market Discounts (retrieved from various merchants in Lestallum, though mostly in Duscae and Cleigne): these quests provide market discounts. A list of these quests and their rewards can be found here.

As far as I know (and as mentioned in Vemonus' answer), the only endless quests are the hunting quests.

Yes, all of those quests have a set number of side quests and all have an overarching storyline that have a definitive ending.

The only quests that are endless/repeatable are the Hunting Quests.