I have 6 industry related regional websites that all link to each other. In fact, each site has a dropdown menu in the header (which appears on all site pages) with 5 links. Should these outbound links be "no follow" links? Is this considered a link wheel?
Absolutely they should be nofollow, From what you have said these links are sitewide so they are generating a lot of links to the target sites. This will make them prime targets to be algorithmically penalized.
If you're doing it to help promote the other sites then maybe you should consider using one of the new tags rel=" sponsored"
you can read more about these here https://webmasters.googleblog.com/2019/09/evolving-nofollow-new-ways-to-identify.html