The simple answer is "not in most cases" - however the total answer is a lot more nuanced.
In the common case where a router is doing NAT, and where the ISP has no login to the router and there are standard protocols in use like HTTP/HTTPS, the ISP can't see the IP address of the individual devices behind the router. A small number of protocols provide this information in an unencrypted form that can be recovered by the ISP. This information can also be leaked by, for example, email headers within email sent through an on-site mail server, or even more rarely by an on site proxy server adding an X-Forwarded-For header for HTTP communication.
It is sometimes possible to trick browsers into revealing this information by intercepting the content. Using HTTPS makes this a lot harder. Its also likely that information about machines behind an IP address can be inferred based on the source ports and other "fingerprinting" information available to an ISP or person tapping into that data. This won't reveal an IP address, but could identify individual machines in the network using the shared IP address - and this can be done regardless of http/https.
All this assumes the router is doing NAT. This is not a safe assumption. For example, if the router has IPV6 enabled, then NAT is not typically used, and the individual device can be seen directly. It is entirely practical (and not uncommon) to have both IPV4 behind NAT and IPV6 available on a router - and IPV6 can be set up without any user configuration or DHCP. Some sites (Google properties for a start) will use IPV6 if its available, even while other sites are using IPV4.
I also observe your statement about ISP logging is an approximation of the truth, but is incorrect in nuance. Most ISP's would not log packets to websites, they would log packets between IP addresses, and very often there is a 1:many or many to many relationship between IP addresses and websites (eg youtube and google have a many:many relationship, and most websites are on shared hosting). Its not shouted from the rooftops, but most governments install equipment at ISP's and siphon off data (there are standardized protocols for this, and IMHO its a scandle that its not well known and publicised in ISP's T&C's). I expect that this government equipment is capable of getting the name of the websites visited and the public IP address of the visitor - even if the data is encrypted (as the domain name is not encrypted by HTTPS in most cases - due to the shared hosting and IPV4 limitations mentioned above)