I don't know why we have to develop a term called "wave"?
Independend from physics, how you want to call the water disturbtion at the sea? And why you call a table a table, although it is made of atoms? Terms are good to describe something, but of course behind this terms are other possibilities to describe things in more detail. A table is a thing made of wood and with 4 legs and a top. As you see you would find tables which are not described well with the detailed definition (3 legs, not made from wood,...).
To me, wave made of particles oscillating up and down periodically. Is wave just a collection of particles or is wave an identity independent of particles?
According to Wikipedia
There are two main types of waves.
Mechanical waves are the waves you described above: Energy transfer without (or only a little) mass transfer in deformable media.
With the electromagnetic radiation it is a little bit more sophisticated.
A light wave is a collection of photons.
Right. Switching on an electric bulb, a stream of photons, emitted from excited electrons, is spread out over the space. And yes, you will be able to measure a swelling intensity of the light. This is due to the AC currency, may be with 50 Hz or with 60 Hz. If you run the bulb with a DC current, it would not be possible for you to measure any characteristics of a wave.
But since humans managed to make radio waves, we are able to conclude from the behavior of such a modulated EM radiation to the photons nature. In an antenna rod a lot of electrons get accelerated back and forth the rod and by this they emit photons. The measurement of the radio waves shows that an electric field component and a magnetic field component, both perpendicular on each other and both perpendicular to the direction of propagation, are produced. This is the sum of the behavior of the photons.
For the near field of the radiation wave it looks like this:

Now you can see why light is associated with a wave.
But,then why does the light still travels in a straight path? If light is a wave then,it should move up and down?
Photons are not moving up and down. Photons are traveling straight forwards as long as they don’t get influenced by magnetic or electric fields in a very specific way. See birefringence:

So the two definitions of a wave for mechanical waves and in electromagnetism is very different and the analogy in EM is due to the oscillation of the electric and the magnetic field components