There are cases where forward and reverse DNS need to match, and you may not want to have any association between two domains. SMTP is something that comes to mind. If your HELO/EHLO domain doesn't match up, there are many servers that just won't deliver your mail. I host my own email, and I work with very technical people. I don't want "fun-domain.com" and "domain-on-my-resume.com" resolving to the same IP address. But I can host them on the same server.

There's still some software out there that does not support SNI.

While your post body focuses on VPS, your question doesn't, so I'll also mention self hosting your own VMs. You can do a lot with reverse proxies and funky port based traffic routers, but sometimes just giving the VM it's own IP is way simpler. Especially if you don't mind hosting the VM, but aren't interested in managing the service. I host a VM for a MUD I used to play. I don't run the MUD, I don't want to. I want them to be able to do stuff on their website without me having to edit a reverse proxy config, or without having to give them access to the host server.

It can also be used to increase the number of connections you can have to a single interface.

Perhaps you're hosting your own VPN and you want traffic to come out an entirely different interface than the one your other services are on, for segregation reasons.

A secondary IP can also allow for a bit of service redundancy. Probably not the most relevant thing in self-hosting land, but the ability to move an IP between two different VPSs (assuming they're on different hypervisors anyway) is pretty handy.