As to the specific question of what does my group use: INKE uses Basecamp. I know CHNM uses Basecamp because the One Week|One Tool team has a slice of that installation. But at least for INKE, that's only one piece of the puzzle, which I'll discuss further on.
I have no idea if that's any sort of standard across institutions or centers, but I do know that Basecamp is popular primarily because of it's ease of use, that it is web-based, and its price. Another bonus is its ecosystem of add-ons/3rd party tools.
That really gets to the heart of the matter: what does "project management" mean to the organization? And, specific to this question, I don't know what a "DH-specific feature" means.
An organization that needs to communicate with its employees and/or stakeholders, about project milestones, to-dos, document storage, etc, yes, a project management suite of some sort -- or at least a plan -- is a good idea. Whether that's roll-your-own using office tools and a wiki or whether that's something integrated into workflow (with actually business-oriented and organizationally-oriented tools like org charts, gantt charts, critical path visualizations, timetracking, so on and so forth)is up to the organization and what the employees will sustain, working within guidelines enforced by management.
I think the Wikipedia comparison of PM software is a good one because the chart itself is categorized well: collaborative tools, issue tracking, scheduling, document management, resource management. Those are all important categories that require careful consideration before selecting a suite that works best for the organization. For instance, if the organization is one that produces software then integration with an issue tracking system may be important -- or it might not, depending on how closely tied the development group is with the management group.
_I_ would not implement any project management system that didn't at the very least have tools for collaboration, documentation management, and scheduling/resource management (with visualization tools). Issue tracking would be next on the list, as there are a whole set of lovely issue tracking tools that can be separated from the management layer (manager of dev group can report to management group at a high level, and keep the details separate). Web-based would be important to me only because I tend to work with disparate teams.
I don't find any of these needs specific to DH. Or, more importantly, not applicable to DH -- they are all applicable, to varying degrees, just as all aspects of management and organization are applicable, to various degrees, to different industry sectors.
Shorter me: INKE uses Basecamp. I like Basecamp. I would also use anything else that met the needs of my team, but most importantly as a manager I would support (and expect) use of the tool as part of employee workflow. It wouldn't be optional.