So I have a personal laptop and coincidentally it's same model as the laptop provided by my employer, so is it appropriate/moral/legal to continue my work with my personal laptop without the knowledge of my employer.
It is wrong to do it without their knowledge. Depending on your industry it may also be illegal.
If they know about it may be okay. Many people check work email on personal devices when they are not in the workplace. Many more do other work on personal devices. Whether this is okay may depend on the type of work you do on your own device. Here are some issues to consider.
Company Policies and Legal Regulations - The company may have policies in place which don’t allow this. These policies may be to secure company systems and information or they may be required for regulatory compliance. You don’t want to break these policies, even unknowingly. Some legal restructions even apply to employees even if the company fail to enfource.
Transfer of Risk - By using your own computer you take on responsibility for guarding the security of the information on you computer. You also take responsibility for any breaches into the companies systems which use your computer as a pathway, for example a virus which infects your computer and then the company system. Even if you have an agreement with the company such as a ‘bring your own device’ policy if you leak company information or your device was involved in a security issue it will may reflect poorly on you.
Separation of Work - Many employers and employees have agreements which define what work belongs to the company and what work belongs to the employee. For example, if you are developing a new product on you own time and with your own equipment then your employment agreement and local laws may ensure that is product is not the property of the company. If you do work for them on your personal laptop then it may be harder to argue which work belongs to them and which belongs to you.
In many cases these issues are far out weighed by the benifits of using a machine which you are very productive using. That is for you and the company to decide.
For many companies, it would be a serious offense.
Imagine you catch a virus or someone hacks your machine - possible because you are not using the security tools the company machine uses, and the hacker edits something into the code you produce. The damage to the company could be astronomical.
As I manager, I would insist that you immediately stop using non-company hardware (unless your policy is different of course), and if you continue to violate it, terminate you.
Ultimately this is a question only your employer can answer, but frankly speaking how would you work on your personal computer?
I’m pretty sure your employer has rules, how/if any company data (documents, source code etc.) can be transferred to any other non-company system. The other thing is, if you use any commercial software for your work, you're probably not allowed to install it on your personal computer.
I don't really see the benefit for using your personal computer. And I would even discourage you from asking your company. Probably the response from your IT department would be "Why would you want to do that?" and might get them suspicious, if you're planning on stealing company data.