How is a PhD project budget set? In experimental fields, such as physics, chemistry, and engineering, every professor has their own lab/workshop, normally with several externally funded projects. A PhD student may work on currently funded projects.

However, it is the responsibility of the university or department  to provide the cost of this PhD project. How and by whom is the budget of a PhD project set?

I mentioned experimental fields, because the costs of these PhD projects are normally much higher than tuition fees (consider that the student has no research fellowship).
In the US and UK, at least, PhD students in he sciences and engineering generally receive funding that covers both tuition and a modest stipend to for living expenses. The exact details of the funding (e.g., taxable or non-taxable, benefits, and whether funding is given for tuition or tuition is waived) vary widely. The funding can come from a number of sources.

The easiest to understand are external grants directly to a student. The grant has a budget and the student is expected to keep the research costs within the budget. Additional costs, potentially including space charges and other overhead type fees, need to either be covered by the student or negotiated directly with the supervisor/department/university.

Grants (both internal and external, and start up funds) to the advisor generally cover a project bigger than a single student. It is the advisor's responsibility to make sure each sub project is appropriately budgeted for and negotiate with the department/university for additional money. While some students will be given an "official" budget by the supervisor, most are required to have the supervisor sign off on all expenses. Budget issues often come up during the design stages of a project.

Departments and universities also fund PhD students. This type of funding is sometimes in the form of Research Assistantships, but more often is for Teaching Assistantships where the student needs to teach in order to get the funding. Research costs can be, but are not always, covered by this type of internal funding. It is left to the student to either cover the research costs themselves or negotiate with supervisor/department/university to make sure the costs are covered.

My experimental research is relatively cheap, but I think in the vast majority of fields, the big cost is staff. Apart from possibly the most expensive types of experimental research, a supervisor with reasonable funding, will likely be able to find the money to cover experimental costs for a PhD student. It might require some limits to be placed on the project (e.g., including some modeling or theoretical work to reduce the costs).