I came across your post here while trying to investigate an issue where aka.ms/mfasetup kept me in a login loop where I would be prompted to enter my username, password, YubiKey PIN, and tap the YubiKey over and over again.
I found that aka.ms/setupsecurityinfo did not do this and I was able to manage my MFA authentication methods. I couldn't find anything else online that suggested there might be something different about these two short URLs. However, I stumbled upon this little note in portal.azure.com -> Azure Active Directory -> Password Reset -> Authentication Methods that suggests /setupsecurityinfo may be newer (green emphasis mine)
While it doesn't answer your question, it at least suggests there is a difference between the two.
As for your actual question, I should warn you that in my experience so far, creating a Conditional Access Policy using the Register security information User Action (under Cloud apps or actions) applies not only to MFA/SSPR registration, but also password resets. So, if you deploy this setting, you will need to make sure all of your users will be able to meet whatever conditions you set for managing MFA AND passwords.