It's about making the differences between species be actual differences of the species, not the culture they come from.
Elves have an innate artistic sense as part of their personalities (which are apparently species-based), and will decorate things like cloth or wood objects in their own culture, but not so much stone. They have the dexterity, the physical and mental capacity to and inclination for such artwork. Put an elf in dwarven culture, though, and why should they suddenly fail to understand engraving just because "stone" or "metal" is somehow something too difficult for an elf to grasp?
The same goes for those tigermen smiths - unless there is some sort of physiological reason they cannot learn how to pound a steel rod into a sword, there is no reason they cannot be as good a smith as a dwarf can be, provided they have the proper training.
Now, if it is something like a druid, and there literally is some reason that a dwarf cannot cast magic the way that an elf can, because some fundamental magical makeup of their being is different, then it's something different from just "because that's what elves are good for". If only elves and forest creatures are born with the nature-magicky-souls needed for nature magic, then it would make sense to say dwarves can either not learn that profession or that they can only advance up to a certain point and even then with great difficulty.
Likewise, if you had an animal-man creature that had no sense of sight, and no concept of aesthetics, then it would make sense to say that they are incapable of almost any form of artwork, not just some specific material they choose to work with.
If it's something like fishing for muscles, and you have a race of water-breathers, then yes, that makes sense to say that the water-breathers are going to be more efficient about collecting muscles, but saying that a lizardman can't learn to dig in the same way as a dwarf does simply because dwarves are stereotypical miners doesn't make sense.
So again, saying just a blanket "dwarves are better axelords" does not make sense - it can make sense to say that dwarves will be stronger than elves, and elves will have more agility, on average, (where the way that attributes are distributed now is perfectly fine) but not that an elf is somehow fundamentally incapable of understanding a concept like "swing an axe" in the same way they understand "swing a sword".
So yes, there's a difference between a physiological difference that provides a strength or weakness, and a totally arbitrary stereotype limitation.