RFC8106 IPv6 Router Advertisement Options for DNS Configuration https://datatracker.ietf.org/doc/html/rfc8106

Ans what software do support it? There are tons of these IPv6 RFCs which is not supported by "common" software - Linux and FreeBSD daemons, nsd, unbound, etc.

How could I implement this in my home network, with off-the-shelf SOHO router or even something like OpenWRT / custom OSS-based "firmware"?

I'm using opnsense and it seems to have worked for my network.

Nice! I last looked at IPv6 in 2019 ish. I wonder how I missed this. Or perhaps the support for it in consumer routers is low?

I guess you haven't heard of RDNSS yet? Doable as part of RAs.

I had not, or had forgotten.

This would be relevant in v6-only networks, but in practice everyone also runs v4 which and v4 dhcp which gets you dns.

(Also there's a DNS option in router advertisements now like subling comment said)

Android ignores any IPv4 DNS address it gets if it gets an IPv6 address. Instead it falls back to Google's own DNS servers. So if you got Android devices in your house, you need to configure a IPv6 DNS server. AFAIK a IPv4 DHCP server can't hand out a IPv6 DNS address.