Interesting point: If the goblins are selectively bred by their owners to be inherently docile to their owners, then by definition they are domesticated. One could argue that there is possibly a domestic animal throughout the world: Goblins.
Obviously enough, without domesticating other animals, you make the acquisition of animal-based products harder. No cows for beef and leather, no pigs for pork and suede, no chicken for meat and feathers. While I'm sure you could cook goblins and they might even be delicious, so long as they are thought of as slaves and not livestock, they won't be.
Meat will have to be hunted for, which will make it rarer and more valuable when it is acquired, as will the related byproducts of the animal. As such, certain instruments might become a luxury as the materials that would go into them are spent on other more important things. Hide for drums, or lyre strings to name two things off-hand. Also of note is that there will be no trained animals to help us hunt in the first place.
Less meat-related: Without domesticated bees, we would not have honey and a ready supply of pollinators for our crops. They would still be pollinated by all the wild insects out there, but we would have less control over that.
Silk is another thing that goes right out the window as the silkworm was domesticated to produce the fabric. Likewise yarn from sheep goes away, limiting materials that clothes are made out of.
As we harvest grain, mice arrive for the free meal. As the mice arrive, things that eat the mice come out and eat them. Things like cats or more accurately, the precursors to the domestic cat.
Without domestication, we would have to rely on nature to provide our initial pest control. Cats that wander to farms and stay are not domesticated, but they might be tamed. From there it is a short step, and a long road, to true domestication. You will have to explain why these other races did not attempt to take that step with the animals that came to them.
Similar to that, the carnivores that hunt the animals that go after our other crops are in a similar situation. If wild canids are coming to us to eat the critters that terrorize our gardens, why did they not domesticate them?
With plentiful slave labour, there is little incentive to innovate automated solutions to problems. Things that do get invented to advance things will either save enough slave lives to be valuable that way, or allow the slave goblins to be more productive.
Lacking large domesticated draft animals, such as oxen and horses, there is the possibility of a reduction in agriculture and merchantile travelling as all carts and plows would have to be pulled by hand or goblin. In this case, a tricycle or similar pedal-powered device might be invented early to aid the goblins.
Other societal issues, both good and bad, will be less related to the lack of domesticated animals and more related to the presence of a plentiful slave race
As one example of that, the lack of domestic animals combined with the goblin slave race to pick up their slack will lead to at least some degree of acceptable speciesism related to them. Depending on the individual, their racial and species-based biases may extend from goblins into other groups that they are not a part of. Alternatively, goblins will not be seen as a group worthy of the same things as the other species. Personally, I would think that an institutionalized and accepted widespread speciesism, no matter what level of it actually exists, to be hindering to a society as a whole.
An interesting effect is that arrows tend to be fletched in feathers. Feathers that are no longer readily available from delicious domesticated turkeys or chickens. While the most likely thing is that they will use a different material for fletching arrows, it might mean that arrows are at a premium in certain areas or are just of inferior quality. Slings, javelins, and spears might be the default ranged weapons just on the grounds of being more readily available.
As leather/suede is not readily available, hide-based armours will not be as available as metal or wooden armour. This is not a deal-breaker, but it could mean that leather armour is a sign of status as opposed to just being light skirmisher-type armour. It may also be that lighter armour is simply a couple of metal parts attached to yourself.
Together, this feels like there is not a lot of ranged piercing power until the firearm is developed enough. Spears can only be thrown so hard, and slings bludgeon more than pierce. Wars could be plausibly based on evasion and not getting hit at all compared to trusting your armour to limit injuries.