HTTPS involves multiple servers that are interchanged regularly and commonly use different keys. The keys change regularly. There'd be no way to know if the key changing is part of regular operation or a man in the middle.
Note that there's a difference between SSH fingerprints and (self-signed) SSL certificates. Multiple servers can easily share a certificate.
It would be unwise to do so. In any case, phasing out old servers and certs for new ones is common practice. Using one cert everywhere (which would mean using one set of keys everywhere, which is a horrible idea) would require more downtime for maintenance of the certificates. It's not gonna happen.