Question: Can you currently install homebrew apps on Windows Phone 'Mango'?
Answer: No, you cannot
One of the firs things we did with the Samung Focus with "Mango" (see review) was to developer-unlock the device using our AppHub account--no problems there. We were also able to sideload official apps from developers not yet released to the Marketplace (e.g. beta testing, early reviews, etc.). However, when it came to side-loading XAP files from say, XDA or other developers, they would not load. Things like Registry editors, themes changers, etc. would result in Error 0×81030120 (see above image).
A few days ago, Rafael Rivera, of ChevronWP7, gave some details about the situation on his blog, which you can read in detail here. [Warning: only developers will understand the lingo]. In short, the technical reason is "Mango won’t officially support the deployment of custom applications with the ID_CAP_INTEROPSERVICES capability flag".
What is not clear at the moment is (a) Is this is a permanent change that will be in the final build of "Mango"? and/or (b) Can there can be a work around for it? (Rivera notes that he hasn't "...figured that out yet, you'll need to standby." in response to developer). It is quite perceivable that Microsoft will keep this as a way to prevent communities like XDA and the homebrew crowd from running Interopservices aka native code on Windows Phones, which will negate a lot of the homebrew communities work. As alternative, developers can release apps to the Beta Marketplace, however even there, running native code (the point of homebrew for many) would not be acceptable there either.
Update: Scratch (a) up above. The final version of "Mango" will have this limitation, according to Rivera, which means the fate of homebrew is up to some ambitious devs.
All we can say is stay tuned.