Do universities typically provide every PhD student with a computer? I worked in some universities in Europe and currently work in Australia. All the universities I have been into supply their PhD students with a PC or a laptop. However, I recently started working with a collaborator in Canada and came to know that his university asks the PhD students to bring their own computers/laptops. 

Are there many universities that do not allocate computers to PhD students? If so, why? and how does it work?
Based on my work experience and from talking to people, I support Bryan's answer. Universities in Europe (not including the UK) virtually universally provide personal IT equipment (typically at least a laptop and a desktop display, plus some peripherals) to every student, while universities outside of Europe, at least often, do not. I do not really know about the UK, but my suspicion is that they may not.

To add to Bryan's answer, I will speculate a bit on the why part of your question.
I think there are is one fundamental underlying reason.

PhD students in Europe are salaried employees of the university first and foremost. This is well documented in many questions here, for example this answer. Every reputable company will provide their employees work equipment rather than expecting employees to bring their own, so it is not surprising that universities do the same for their research assistants / PhD students. Contrary, master's students (who are usually not employees in Europe) basically never get a laptop to work on their master's theses, and are instead asked to use their private equipment or the public computer rooms of the university.

Along with this long-standing tradition to see PhD students as employees also comes a different mindset. Universities in Europe typically assume that all resources required to conduct the PhD project should be provided by the university. After all, we don't expect students to pay for trips to conferences, open access charges, or time on expensive experimental hardware out of their own pocket, so why would we assume that they pay for their basic work equipment?

Finally, it's also a question of basic economy. In Sweden where I currently teach, a PhD student costs me more than 350k USD over 5 years (the normal PhD duration in my department) in salary, social security, benefits, and overheads. Spending another 3k to provide the student with good hardware to work with really does not make a whole lot of difference in the overall project budget, while contributing quite positively to the overall PhD experience and likelihood of success for the project (or at least we think it does).