Yes, you could do so through the tech tree. It's possible to restrict techs so that they only appear for a given race using the following syntax which I copied from an older 5X version:
In the above code, toxic condenser only shows up in the tech tree for Meklar. The spacing I used must be copied exactly for it to work. At one point I was considering creating a unique tech tree for each race where everyone gets their own unique weapons. In the end I decided it would be too massive of an undertaking.

I'm not sure how the above code works in a "branches" section, so you'd have to experiment with that.

Unfortunately, this only works for stock races. Custom races can't be given restricted techs in this way. For custom races, if you wanted a certain weapon to unlock when researching biotechnology for example, you could create a race perk that unlocks a hidden tech node which requires biotechnology so that once a race researches that, the hidden node is unlocked for them. Unfortunately that tech wouldn't show up in the tech tree. You could use a dummy tech as a stand in, but then that dummy tech would appear for every race (this is basically what I ended up doing with toxic condensers in 5X). You could of course use this method for stock races as well if you gave them the race trait you created.

Also, unfortunately free tech perks can only unlock one tech node at a time, but you could fix that by creating a combo perk with all the free unlock perks you want to include.