There are several qualities in the game that relate to specific weapon categories, most notably the "[weapon category] Fighter I-III" line. Some of these are pretty clear in that their benefits apply to only this weapon, such as Short Blade Fighter I adding Piercing 1 to the weapon or Polearm Fighter I stating "you can sweep your pole-arm". Others are pretty clear in that they don't - Water Dancer I and II require points in Fencing, but grant passive benefits that are written to be universally active.
Some don't mention anything of the sort, but can be argued to be self-evident. Spear Fighter I doesn't mention spears at all, and by RAW probably could be used with any weapon, but I would be receptive to the argument that the quality is meant for use with spears only. The flavour text supports this.
Where I run into trouble is the qualities that require you to have a weapon equipped, but don't explicitly require its use. If I look at Bravoosi Fighter III for example, it requires the character to have a fencing weapon equipped. However, imagine someone using a left-hand-dagger (a fencing weapon) in one hand and another one-handed weapon, maybe a battleaxe or longsword*, in the other. It could be well argued that the character is "armed" with a fencing weapon and thus fulfills the requirements to use this quality, but could make the counterattack with the other (more damaging) weapon. It's not even implausible in terms of fencing, on the contrary, using one weapon to parry and the other to counter is perfectly sensible. Several other such qualities can lead to similar combinations.
I assume that the RAW here is just sloppily written, but do we have any source on what the RAI is and what are the balance considerations towards allowing or disallowing such combos?
*in the sense of an arming sword that the game calls a longsword.