Is it unusual for a math department not to have a mail/web server?

I've found out that the department (or the university?) is doing away with the server and instructing everyone to instead use the G Suite email and Google services that the general student body of the university uses.

It sounds like the university is outsourcing IT, which is normal.

It is becoming less common for individual departments to operate their own IT infrastructure - things are usually being centralised.

It is also becoming more common for parts of that centralised IT infrastructure to be outsourced - which in practice means that email ends up on Gmail or Microsoft's system. I would expect that your email address will still have uniname.edu as its domain - it will just be operated by a cloud provider.

Along with this second tendency, it's becoming less common for universities (or departments) to provide web space to staff or students. This can be annoying. But as others have said, there's nothing unprofessional about setting up your own domain (with a sensible name) and hosting your web content there.

I'm worried that having a Google Site as opposed to a .edu website will appear unprofessional...

I think that this worry is unfounded. Similarly, I don't think that it would be unprofessional were you to have a url like www.MikePierce.com. You just need to avoid having to put something like www.badboymowers.com/mikepierce on your CV when you apply for jobs.

More seriously, several of my collaborators use Google Site pages as their main professional websites, and I know of several more respected mathematicians that do so as well. And this is all off the top of my head. So I doubt you'll stand out in a negative manner if you have to migrate your website to a Google Site.

Similarly, there are also a lot of people that have .edu email addresses but choose not to use them professionally (though this is much less common). For example, one of my collaborators has a website hosted by his institution but lists his gmail email address in all of his publications.

Should I be upset by this? Is this unusual? Or is this just the trend of the future? Is this more typical than I think it is?

I think that your department's situation is that of a growing minority of departments. I think that if you have a professional looking Google Site page that is linked to by your department's official website then you'll have nothing to worry about.