When the word synthesizer is used, it's not meant to be a piano like thing or? For me the word is analogous to "piano keyboard" but it seems to not be so...

A synth is a device that generates sounds. They’re often (but not always!) controlled by piano-style keyboards, but so are lots of non-synth things. For example, although a piano emulator could be a synth, they’re generally not thought of that way. A sampling keyboard is not a synth: it’s playing back recorded sounds.

Lots of synths spent their whole lives never connected to a keyboard, but might live in rack mounts and be driven by sequencers or DAWs.

Some synths have piano keyboards attached to the device, some have buttons. Some piano keyboards are not capable of producing their own sounds, and must be plugged into a device that makes the sound.

A synthesizer is a device that generates (synthesizes) sound. This is done using electronic circuits. There are many many many many different ways to create sound using electronic circuits, hence the many different types of synths on the market. These days however, we see more and more synthesizers using digital circuits or even embedded software to generate sound. It has reached a point where it's possible to take the firmware of a (very expensive) "hardware" synthesizer, run it on your computer inside an emulator and get exactly the same sound out of it. Heck, some commercial synths (Korg) are these days nothing more than a box with knobs and a raspberry pi, running their commercial synth plugin software.

The "piano keyboard" in your mental model is merely the controller - the device that generates the instructions on what note the synth must play. Piano keyboards are not the only type of control mechanism available to use. For every type of instrument out there, there probably is a controller version of that instrument, e.g. https://www.akaipro.com/ewi-usb

The OG synthesizer concept is the combination of oscillator + frequency filter + amplifier.

As a musical instrument it is closest in function and mode of operation to the singing human voice and nothing at all like a piano.