I don’t want to go so far as to tell you how to think, but as long as we are talking about how to visualize IP addresses, you may want to check out subnets and subnet masking.

The notation of IP addresses starts to make sense when you think about the early days of TCP/IP when all IP addresses were public and NAT’ing wasn’t really required yet. Basically, there needed to be ways for networks to filter traffic by IP blocks that were applicable. (It was [in part] a precursor to collision avoidance, but absolutely not the full story.) We still use addressing and masking today, but it’s more obvious when it’s local. (Like in data centers, where it’s super practical to mask off a block of addresses for a row or rack of servers.)

To your point, yeah. IP addresses are probably more comparable to the Dewey Decimal System rather than actual numbers and thinking of them as strings is probably easier.