I think you're both thinking in the right direction, but you're asking different questions. I think the first thing to talk about is that bray curtis distance is "beta diversity" but it's a measurement of distance (you can learn more about that in a couple of posts I'll link at the bottom).
So, your boxplot measures the distance from the CON group to the CON, DFM, TD, and TYL groups. When I look at the boxplot, the distance between the TYL group and the CON group has a wider spread (a bigger box) than between the other groups, but the mean or median (I dont remember which it is in this boxplot) is lower. So, I would read that the TYL group is more similar to the CON group than the CON group is to itself, the DFM group is to the CON group or the TD group is to the CON group. You can test the that difference in the means (the group similarity) using a Permanova. In that, your PI is correct.
However, the variance is a different thing to test, and that can tell you something different. It looks like the variance in your TYL group might be larger (based on the wider spread you have in the interquartile region). You can test that with a permdisp, which asks if the dispersion (kind of like the variance) in groups are different. (There's another post about that I'll link below as well).
Finally, one thing I always thinks help is to remind yourself about what different things mean in a boxplot. I like this resource, your milage may vary.