songs sound...

In general, if two songs sound strikingly similar, the odds are they are based on the same set of notes and thus in the same Ragam. Their basic Ragam is identified typically by pattern recognition, if you are not willing to do detailed decomposition into the constituent keys of their scale.
The basis of Ragams is (1) the use of a restrictive number of keys in an octave (2) go up and down in the octave in a prescribed manner. (3) And yes, throw in the appropriate microtones. These generate specific melodious personalities. The term microtones presents a major difficulty in understanding the totality of the concept 'Ragam'. How exactly can one specify which microtones are involved ? What is the best way to 'notate' the millions of intermediate frequencies ? Instead of getting very analytical about 'microtones' Karnatic music just gets away by omitting a precise definition of a Ragam. In some sense, the 'Arohanam'-'Avarohanam' - this is the ascending sequence and this is the descending order defintion of a Ragam is only an 'operational definition' at best. Since the 'associated microtones' or 'Gamakam' cannot be defined numerically, it has also become fashionable to simply say that a Ragam is a 'Mood' or a feeling or an emotion, if you can even relate to such unmusical terms.
Another way to define a Ragam is by analogy or how it should 'sound' like. And compare it with an established historical 'standard' or 'primitives'. It is always much easier to sing the 'Gamakams' associated with the Ragam - produce the basic patterns - rather than Fourier analyze it. A ragam is alternatively defined in terms of its 'characteristic musical phrases'. These characteristic phrases are called 'Pakads', (in Hindustani music) literally meaning 'catch' phrases.
All these lend a certain amount of mystery to the concept of Ragam. Like blind men trying to figure out an elephant, we are supposed to only know 'a part of the personality' of a ragam. Never its wholeness. We can only know so many 'characteristic phrases' of a ragam, not a complete set of them, even if there exists such a complete set. One song may have twenty of them. Another song in the same ragam might use just ten of them, but a ten other new ones. Musicians are always trying to create newer and newer 'characteristic phrases' to bring out newer and newer aspects of the ragam. One might have thought that they would have composed every possible phrase in the Ragam Shankarabharanam. But people are still making new melodies in this centuries old Ragam ! We will perhaps never run out of tunes in this Ragam.
The easiest way to identify a ragam then is by analogy and trying to figure out if there is a recurring characteristic phrase. Figuring out a Ragam has always been a natural thing for a Karnatic music enthusiast, especially if the Ragam happens to be an obscure one. It is almost like solving a crime. Some of the ragams can be so distinctive that you can recognize them when only two or three notes are played, thanks to the characteristic microtones.
Sometimes, life is not quite simple. Our definitions of the term Ragam may be violated. Some talented musicians might introduce extraneous notes occassionally into a well-defined ragam, for nice musical effect. Such a process is called 'Misra'fying. You can have a ragam Sivaranjani played pure - this is a pentatonic ragam. Or you can have Misra Sivaranjani where you occassionally introduce a sixth or seventh note not prescribed in the definition of the ragam. Note that this requires expertise. If you or I play Sivaranjani and try to Misra-fy it, we may go so far off the original ragam that we might sound horrible - resulting in 'besur' or 'Abaswaram'.
Equally incredibly, we have other violations as well. Ragams like Sindhu Bhairavi and Kapi are often played with many more notes than just the traditional maximum of seven. On the low side, people have laid claims to Ragams with just four notes. Again, let your ears be the judge.
Play some classical sounding music and try to see if any particular Ragam thrills you. Anything that turns you off completely ? Play instrumental or light classical music at first before embarking on a heavy-duty vocal piece. Is there a piece that moves you? Puts you in a sublime mood? Helps you drive your car ? Goes well in the background when you cook?
The reason for asking these questions is to figure out a little bit about the psychoacoustics. While I do not believe that a particular Ragam could inherently be an 'Angry' Ragam or a Midnight Ragam or bring the rains or tame an elephant, Ragams could very well produce individual psycho-acoustical effects.