Do I need iCloud storage for Time Machine backups on a Mac without other Apple devices?

No.

iCloud is great for syncing files for access on multiple devices and for backing up those mobile devices. iCloud isn't designed for backing up Macs, so Time Machine is all you need for that.

If you don't have other devices, then you really don't need iCloud at all.

I'm trying to think of what value it might have for you. You didn't tell us what sort of Mac you have.

iCloud is a synchronization service, so its primary value is in keeping pictures and documents the same on all your devices. Clearly that's not important in your case.

You have Time Machine which is great, but most of us would think that's not enough. I have two external drives running Time Machine-- one at home and one that I carry around with my MacBook. I also have separate drives devoted to picture backups. Generally it's advisable to have backup that won't be destroyed if your house burns down, so something should be kept off-site.

So, you need a minimum of two backups, and iCloud might provide the off-site backup for you. iCloud by itself is a lousy backup, but it can work well as an augment to a Time Machine backup.