Do companies prefer unpaid experiences that match the skillset/experience requir...

I am a very recent MaSc grad in Aerospac Engineering in Canada. I have a Co-op internship during the 2019-2020 (that got cut short because of COVID laid off). I have been volunteering in an aviation museum to restore aircraft and my main duty is using my engineering knowledge to design and analyze stuff for them. I do this because I want to get more hands-on experience (can be proven with the engineering drawing diagrams and analysis software files). Now I started applying for entry-level engineering work and haven't heard any replies, I start to get worry about whether my volunteer work can actually help or the companies care more about paid-job experiences, even if it is just dishwashing or as a cash register.

Every position can carry information for an employer. It's helpful for you to demonstrate that to them. So yes, your volunteer work can show that you're using your engineering skills, which is great. Another candidate might use their unrelated paid work to show their work ethic, punctuality, and ability to get along with coworkers. You can't fix that by worrying. What you can do is make sure that your unpaid work demonstrates your work ethic, punctuality, and ability to get along with coworkers. Have stories ready to tell. Make it clear in your resume whether this position had fixed hours or deadlines, who you worked with, and how you were evaluated.

I am sure there are volunteer positions where the volunteer showed up when they felt like it, worked until they felt like leaving, and didn't make a big impact overall. If you're worried an interviewer might think that's what you did, you need to tell a story (on your resume, in your cover letter, and at the interview) that sets the record straight. Show them that you are a great worker in your volunteer position, and you won't worry about competition from great workers in unrelated jobs.