I'm not sure what you mean by grammatical gender. If you're talking about the way e.g. German has der, das or die for all nouns, where der is masculine and die is feminine, i.e. implicitly giving all objects a gender, then Swedish doesn't have it. We have two grammatical genders, den and det, but neither is either masculine or feminine. So a table is just "it", we just have to types of "it" (and the way we divvy up nouns between them seems pretty arbitrary).
But I can tell you about pronouns!
We do have a lively discussion within Swedish RPG design, and Swedish language generally, about "he/she" and similar constructs; third person singular where gender doesn't matter, isn't known, or doesn't fit into the gender binary. In traditional Swedish written language, there's han (he) and hon (she), and if you want to remove gender you'll have to do (sometimes awkward) reformulations like "the person in question" or "the player" or whatever. But there is the gender-neutral pronoun "hen", first suggested in the 60's but gaining traction in the 2010's. However, there has been an online debate, much more polarizing than one would've hoped was necessary, with "hen" opponents calling it unnecessary or unnatural ("let's face it, every person is either male or female"). Obviously it's both a useful word (for contexts where gender really is unnecessary), a good word (allows non-binary people to assert their identity and have it respected), and a real word (people are using it, so it's a "real" word whatever you happen to think about it).
However, because of the venom spewed on "hen", a lot of people shy away from using it in formal contexts, and it's gained a sort of... visibility. It's like, it has the potential to be this very compact, utilitarian tool in RPG rule text, where you always talk about player A doing something to player B and then the game master does this other thing, and at no point does it have any relevance whatsoever who among them identify using he, she, or otherwise. But if you write a text like this, everyone's first reaction is going to be "wow, they are really using 'hen' a lot in this text" – even readers who themselves support the use of "hen"! It's become noteworthy, and stands out, and will probably have to be eased into written language for quite a while more before it feels natural to read. Which really is a shame. Why the fuck should it be so hard to consider a person divorced from their gender? (Spoiler: because of patriarchy and the gender binary.)
There are similar gender-neutral pronouns in English (e.g. Spivak) but they aren't, in my understanding as common or heavily pushed as "hen" in Swedish. In turn, you have the singular they (with a similar debate, of course). Sadly, it doesn't really work as well in Swedish as in English, probably only due to exposure – it still "sounds wrong" to too many people.
Now, the original AW has decided grammatical genders on most or all of the characters who show up in examples. I think all players and MCs in the examples, and the general "the player" and "the MC", use "she". We're considering different options on how to translate this. We really want to use "hen" – it's sad that this is the case, but using it in a printed book is a political action; one that all of us at Skepnad Studios support; and one that we also feel aligns with the spirit of the game itself. At the same time, we don't want it to get in the way of readability or text flow, so we don't want to overuse it. Bakers' choice to have irrelevant genders consistently be "she" deserves some sort of respect, but I'm also confident in saying that using "hen" won't clash with their intentions either.
After writing all of this, I realised your question might have been prompted by the Tigress, since it's gendered while "the Tiger" would read as ungendered. When it comes to animals, professions and probably a number of other categories of words I can't recall right now, yes, we usually have one male and one female version. Lärare and lärarinna mean teacher and teacheress respectively, for example. However, the evolution of Swedish for quite a long time has been to stop using the female form, and consider the male form universal. So "lärare", while originally/technically male, can be used about a female teacher no problem. For a few exceptions, like "sjuksköterska" (nurse), the female form has become the accepted gender-neutral one.
This means that if you want to construct a playbook name from a verb, like "driver", you would use the "technically male" variant, and no-one would bat an eye. For "förare" (driver), like many others, you could construct the feminine "förarinna" if you really wanted to, but it would just sound archaic or made-up. So other than the battlebabe, all playbook names are nouns or "verb-ers" that are essentially gender-neutral. With the battlebabe though, I really like the fact that the English name is gendered, and would like to preserve this property. And while "tiger" (same word in Swedish) would read as gender-neutral, "tigrinna" (tigress) does have different connotations. There are associations not just to the jungle but to amazons; not just to felines but to cat burglars, Catwoman. A tiger is dangerous, a tigress is dangerous and sexy. It's not perfect, and I'm still unsure on whether it's the best choice, but I do like it. It being gendered, though, is a deliberate choice, and not a limitation of Swedish (more than the fact that "babe" reads as feminine is a limitation of English).
I hope I'm making sense, despite using all these Swedish examples that probably look like, heh, Greek to some of you.