Is it okay to refuse a laptop from a new university if you already have your own personal laptop?
It surely matters where you are.
In the U.S., at many universities, "privileged information" of various sorts is supposedly not ever to be kept on "personal", as opposed to "institution-owned and maintained" machines. Or, as in some comments, as soon as you do have work-related data on your personal machine, that machine becomes liable to Freedom-of-Information warrants, and you yourself can get into various sorts of trouble for insufficiently guaranteeing security.
A similar issue exists (depending on jurisdiction, etc) with regard to email accounts. My (U.S. R1) university account is subject to search without too much probable cause. Some of my colleagues hesitate to have any substantive-sensitive discussion by email because the University's policy is that we are not to delete any such email, but preserve it indefinitely. That kind of thing. (No, it's not clear how this would be enforced, nor what the impact of "pleading ignorance/technical-incompetence" would result in. Maybe it's just CYA policy on the part of the institution.)
First, I would strongly suggest that you have a work computer and a personal computer, and then keep those two separate for legal reasons. Although this is not the place for legal advice, and there are many other factors to consider, you should know that in general:
Your employer owns your work computer, and can legally confiscate it at any time and for any reason. Thus, you should consider any personal information you have on your work computer to be accessible by your employer. This includes personal information like tax forms and private correspondence. It also includes information you might not want your employer to have, like criticisms of the administration or job offers from other institutions.
Academics tend to have many varied endeavors inside and outside of their academic profession. Your employer probably has a very strong claim to the intellectual property rights of anything you create on their computer, even if the IP does not relate to your university job and even if you're only using generic software such as Microsoft Word.
Your position at a university may expose you to FERPA or HIPAA protected information, and your university may have specific expectations about how you access that data. My university insists that all laptops use whole-disk encryption because the loss or theft of unencrypted student records is a major FERPA event that must be disclosed to the government and/or public.
Second, there are some practical and legal problems with retaining your own computer from a software licensing point of view.
There's a high probability that a lot of the software on your current computer should no longer be used according to common academic licensing agreements. This is definitely true for certain specific software such as MATLAB, which are generally licensed to the university for use by university students and employees (this is called a "site license"). Since you are no longer a student of that university the license demands that you stop using that software.
This all depends on the specific licenses that your university has negotiated, but the situation above is very common. It almost certainly applies to any proprietary technical software you've got, and there's a very good chance that it also applies to any proprietary productivity software you've got (e.g. the Microsoft Office suite, VPN software, etc.). It may in some cases apply to the operating system itself, though this is less common today than it used to be.
The reverse of the above situation is also a problem. Many university licenses stipulate that software may only be installed on university owned computers or on student computers. As a postdoc you're no longer a student, so their IT department might balk at installing any work-related software on your personal computer.
It's probably best to write to them saying you would prefer to us your own computer and ask whether that's an option.
If they do require you to use their machine, you could order one with very similar or compatible hardware to yours, image the disk of your old laptop to an external hard drive, and restore on the lab computer. No installation or configuration necessary.