It depends. Some of the more famous, bigger names can have a sort of theme to their work - eg very dense, heavy scents or very light airy scents or a particular way that the composition evolves.
A perfume house with an in-house perfumer may more or less base their scents around a certain core in order to create a house 'feel'. If they use more than one perfumer that 'feel' can be part of the brief. If it isn't a perfume house but something else - a store, a brand, a celebrity, an occasion or whatever, then yes, the perfumer will almost always get a brief. It would usually be one of the bigger names that would be given a free hand - that is, no particular brief. It would be unusual but it happens.
I don't think any perfumer could be a perfumer if they always produced the same sort of fragrance. I have hundreds of perfumes - been collecting them for all my life - and I will sometimes spend a month or two wearing scents by a certain perfumer, from their first to their last for example. It doesn't follow that I will like them all though.
Some perfumers can manage to get the funding to start their own perfume brand which is one of the reasons that there are so many niche houses these days. This is where in theory a perfumer can operate without a brief. IMO not all of these smaller, niche houses are as good as they think they are (and some are frankly awful), and the problem for the few really good ones is that they are lost in a sea of marketing, social media and paid-for fashion trends.
In fact the one thing that I actually cannot get my head around is that people blind-buy scents based on the recommendation of someone else who may have a completely different taste, or some subsidised 'influencer' or even based on reading Fragrantica or whatever (it is worth keeping in mind that many fragrance reviews are clearly from people wearing dupes or eBay fakes for example! Or that they are drawing a bizarre comparison with the one and only other fragrance that they own.) It is so worth the time to go and try scents and find one that you love just as you would with an expensive item of jewellery or a good dress.