The answer is yes, the serial port is obsolete but it's still needed, especially for Linux. The serial port has many shortcomings but almost all new PC's seem to come with them them. Linux supports ordinary telephone modems only if they work thru a serial port.
The serial port must pass data between the computer and the external cable. Thus it has two interfaces and both of these interfaces are slow. First we'll consider the interface via external cable to the outside world.
The conventional EIA-232 serial port is inherently low speed and is severely limited in distance. Ads often read "high speed" but it can only work at "high speed" over very short distances such as to a modem located right next to the computer. Compared to a network card, even this "high speed" is low speed. All of the serial cable wires use a common ground return wire so that twisted-pair technology (needed for high speeds) can't be used without additional hardware. More modern interfaces for serial ports exist but they are not very popular. See Successors to EIA-232.
It is somewhat tragic that the RS-232 standard from 1969 did not use twisted pair technology which could operate about a hundred times faster. Twisted pairs have been used in telephone cables since the late 1800's. In 1888 (over 110 years ago) the "Cable Conference" reported its support of twisted-pair (for telephone systems) and pointed out its advantages. But over 80 years after this approval by the "Cable Conference", RS-232 failed to utilize it. Since RS-232 was originally designed for connecting a terminal to a low speed modem located nearby, the need for high speed and longer distance transmission was apparently not recognized.
To communicate with the computer, any I/O device needs to have an address so that the computer can write to it and read from it. For this purpose many I/O devices (such as serial ports) use a special type of address known as an I/O addresses (sometimes called an I/O port). It's actually a range of addresses and the lower address in this range is the base address. If someone only says (or writes) "address" it likely really means "base address"
Instead of using I/O, addresses some (non-serial) I/O devices read and write directly from/to main memory. This provides more bandwidth since the serial I/O system only moves a byte at a time. There are various ways to read/write directly to main memory. One way is called shared memory I/O (where the shared memory is usually on the same card as the I/O device). Other methods are DMA (direct memory access) on the ISA bus and what is about the same as DMA (only much faster): "bus mastering" on the PCI bus. These methods are a lot faster than those used for the serial port. Thus the conventional serial port with it's interrupt driven (every 14 bytes) interface and single bytes transfers on a bus which could accommodate 4 (or 8) bytes at a time is not suited for very high speed I/O.