The fonts are the same designer... it's just that Monotype owns the Font Bureau fonts now. As far as having both loaded at the same time, it definitely creates a conflict, and they tend to be handled on a "first come" basis in terms of how they are loaded in your System. Even so, this can happen differently on a different computer or at different times.. Plus the font caches can get messed up. Despite being different formats, the internal Postscript Name is the same, e.g. "AgencyFB-Bold", so InDesign sees that and says, okay, got it! So despite you loading the fonts from Adobe Fonts, the Bold weight is ignored as it's seen as "already there" and won't allow both to be loaded. This is a System level thing, not InDesign.
It may not be enough to deactivate the TT versions. Once you have done that and confirmed the Adobe Cloud versions are loaded, you may have to literally do a find and replace in your document to replace the "missing" AgencyFB-Bold with the "loaded" AgencyFB-Bold for it to clear up (making sure to modify styles at the same time).
Like most people that use a lot of fonts, I use a Font Manager which makes it helpul to see what fonts are loaded at any time. In my case here, I have AgencyFB Bold loaded already from Adobe Cloud Fonts, but it will not let me activate the TrueType version stored locally on my computer at the same time. If, howver, i had loaded the Adobe Fonts AFTER starting with the TT version in my system, it would load all the Adobe weights EXCEPT Bold.
 
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