A, AAAA, www, and @ are DNS labels. DNS is the list that turns a name into an address. Search Console does not write that list, except for one TXT row when you verify a Domain property. The rest of the rows decide which computer answers when someone opens the name.
@ is the domain itself
RFC 1035 says a free-standing @ in a zone file denotes the current origin. That origin is the domain the file is for, such as example.com. Google's ownership page uses the same mark on a verification TXT record: leave Host/Name blank, or set it to @. That row is for the domain, not for www.
www is a second name
www.example.com is a different owner name from example.com. RFC 1034 says a CNAME identifies its owner as an alias and names the canonical name. If a CNAME is present at a node, no other data should be present. The name can be an alias, or it can hold address records. Not both.
Search Console does not merge those names unless you add a Domain property. The add-property page says a URL-prefix property includes only URLs with that prefix. http://example.com/ does not include http://www.example.com/. A Domain property for example.com includes www, m, http, and https. If you type www.example.com as the Domain property, Search Console creates it as example.com. Each extra subdomain still needs its own URL-prefix property when you want that slice alone.
example.com covers www. https://example.com/ does not.
A is the IPv4 address
RFC 1035, section 3.4.1, defines an A record as a 32-bit Internet address, written as four decimal numbers separated by dots. A host with more than one address has more than one A record. @ in the zone file is the origin. www is a different name, so it needs its own A records or a CNAME, not both.
AAAA is the IPv6 address
RFC 3596 defines the AAAA record, four A's, type 28. It stores one 128-bit IPv6 address. A host with more than one IPv6 address has more than one AAAA record. The same document keeps the existing IPv4 records, so one name can have both A and AAAA. A CNAME at that name is still forbidden by the RFC 1034 rule above.
A zone apex already has an SOA record. A CNAME there would share the node with other data, which RFC 1034 says should not happen. Cloudflare documents a product exception: it accepts a CNAME at the apex and answers with the address instead of the CNAME. That is Cloudflare's behavior, not the DNS standard. A provider without that feature rejects a CNAME on @.
MX, TXT, and NS
RFC 1035 defines the other three. An MX record names a mail exchange and a preference. Lower preference values are preferred. It does not choose the website. An NS record names a host that should be authoritative for the domain. A TXT record holds text. The meaning of that text depends on the domain where it is found.
TXT is the row Search Console uses for a Domain property. Google says DNS verification is the only method for that property type, and that most sites use a TXT record. The value looks like google-site-verification= followed by the string Search Console gives you. The host is blank or @. The other DNS option is a CNAME whose value includes dv.googlehosted.com. Google's wizard tells you which of the two to add.
What to change, and what to leave
Google says adding a property does not affect the site on Google Search. It only lets you track it. Leave the verification TXT in place. Verification lasts while Search Console can still find that token. If both example.com and www.example.com get visits, one Domain property already includes both. Two URL-prefix properties stay separate, one per host and protocol.