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My understanding is there are 2^32 - 1 possible IPv4 addresses, and 2^16 - 1 possible ports. Which gives ~2^48 addresses.
The additional 2^16 additional ports seem almost insignificant considering the IPv6 address space is 2^80 times larger than the number of IPv4 addresses * ports. With the 2^128 possible IPv6 addresses, why do we need ports at all?
Why not assign each application, tab, etc... its own public IPv6 address?
An IP address targets a host on the network layer. Transport layer ports multiplex an L4 protocol within a host (to different processes/services). Both are different things on different layers.
Basically, if you'd repurpose IPv6 addresses (or bits) for host-level multiplexing you'd gain very little but you'd break logic compatibility between IPv4 and IPv6. A transport layer protocol works the same way on any IP version.
Why do we need ports with IPv6?
As Zac67 already said, it was intended to make IPv6 as compatible to IPv4 as possible to ensure that existing programs need not to be changed too much.
Keep in mind, that in IPv4, the destination port of a TCP connection also specifies the type of higher-layer protocol:
80 for HTTP, 443 for HTTPS, 23 for Telnet and so on.
So we need the TCP port in IPv6 as indication of the higher-layer protocol type if we want to be compatible to IPv4.
Why not assign each application, tab, etc... its own public IPv6 address
I was already thinking about this: Doing this would allow implementing certain additional security and privacy features. So I can imagine that this will actually be done in the future.
And of course: We can get rid of URLs like https://example:8443/: Instead of using different ports (8443) on server side, we can simply use multiple IPv6 addresses.
However, it would still not allow us to get rid of the source port of a TCP connection:
I rembember some web-based application that requires two HTTP(S) connections to be established between the client and the server at the same time.
In this case, the server will surely check if both connections come from the same client (this means: same source IP address).
However, this means that both connections have the same two IP addresses and the same destination port (443 in the case of HTTPS). Therefore, they must have a different source port.
And if IPv6 should be compatible with IPv4, this web-based application must work the same way with IPv6 as it works with IPv4.
I think the OP is not so much saying that we should roll ports specifically into the routable address scheme of the internet, but instead why not give every application/process/thread its own unique, routable address?
It's definitely a valid question, and you're not the first person to wonder this - to quote John Day "Networking is IPC and ONLY IPC".
Currently, this kind of thing has to be handled at the application layer - if I'm visiting Facebook on my phone over cell signal and then my phone switches over to WiFi, technically my source IP address (and port) has changed completely from Facebook's perspective, and it's only the tokens passed by my browser that are accepted as being the same (application-level) session.
But why stop at doing away with ports? Consider the situation with phone numbers - a phone number is just a poor proxy for specifying the identity of the person you want to speak to. That's why phones have contact lists - phone numbers are just a leaky bit of implementation that we want to ignore as quickly as possible. It's why DNS was invented - the authors of the Internet protocols chose to implement a centralized/hierarchical naming authority system versus your phone's local contact list, but same idea.
What you will want to read about are alternative Internet protocol proposals like John Day's Recursive Internetwork Architecture (RINA), where he critiques several historically accumulated design flaws in TCP/IP and the rest of the Internet protocol stack and proposes RINA as a solution.
Specifically to your question, a couple of bullet points out of John Day's critiques are relevant:
Whether you agree with him or not (as far as I know this proposal doesn't have a ton of traction), the flaws he points to and solutions he proposes are quite thought provoking.
You need something to say "this bit of data coming in from the NIC needs to go to this process."
This is different than "which host/network does this data need to go".
IP addresses are processed by your router and potentially other carrier grade routers to determine where to move your data.
Those devices do not have the ability to, for example, reach into your PC and deposit the network data directly into the process's RAM, so they don't need to care which process on the destination wants it.
This is like adding "Give this to your cat at 2pm" as part of the address of a package you want delivered. The people processing the package don't really need to know that, they just need the address, so it's better if you put those instructions inside the package.
Currently in IPv4, the number of valid ports is 65,535 - but 0 - 1023 are restricted or reserved port numbers (Often reserved by specific company processes as well - for example, apparently Apple QuickTime, and Remote Desktop Protocol does.
Natural Address Translation takes advantage of the fact that, for most users, which unrestricted and unreserved port being used generally doesn't matter for the average use case, and doesn't need to be aligned between client and server, so we can re-map them. But what made it required to be a thing that we went with was that IP address blocks were bought by companies in large, "We'll never run out of numbers to assign!", well...numbers. Large enough to run out.
On top of this, once machines move around and switch ISPs, then they would run through more IP addresses, and their previous one would have to be expired when routing to them.
With a single IP address per machine, you can "keep" that old IP address assigned to that older device, even if it temporarily leaves where it was assigned the IP address from for a short time (More likely to be a mobile machine, or a laptop), and keeping those lower will help.
Granted, 2^128 is a lot of IP addresses, but ports will allow us to avoid running into a similar version of the problem with IPv4 thinking it had more than enough IP addresses for everyone and their dog.
0, 1023, 1024, 49151, 49152, and 65535.