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I am a beginner in music with little knowledge, but like once in a month spend a little time playing with a digital keyboard.

I noticed that if I have some notes of a song which got 4# in the beginning of the stave, I can play the song like there is 3♭ instead. The same happens if I have 4 bemols (flats) then I can play the song like it has 3 diesis (sharps) (yes, it will sound a bit higher, but not that different).

I was curious if other combinations exists, let's say we have a song in 2 bemols (flats), what is the equivalence of it in diesis (sharps)? I couldn't find it myself. Is there a name for this phenomenon so I can learn more?

I'm not aware of a name for this phenomenon, it's just a quick way to transpose music based on how the tonal system works out.

In short, when you're in a key, look at the key signature. Take the number of accidentals in the key and replace them with the mod-7 complement of the other accidental type and you're left with a key built a half step away from the original tonic.

So you're in E major with 4 sharps. Let's take the mod-7 complement of the other accidental type: 7-4=3, so we're left with 3 flats, which is E♭ major, one half step away from the original tonic of E.

You're now asking about 2 flats in the key signature; this is B♭ major. 7-2=5, so a key of 5 sharps will be B major.

This trick is especially fun in C, which has 0 sharps or flats. The mod-7 complement of 0 is 7, so if we have 7 sharps in the key signature, we're in C♯ major; 7 flats makes it C♭ major!

Note that this trick isn't exclusive to major; it works for minor keys as well.

Lastly, know that this works perfectly until you encounter accidentals in the music; you'll have to have a more contextual understanding of those accidentals to know how they should be interpreted in your new key.

I was curious if other combinations exists, let's say we have a song in 2 bemols (flats), what is the equivalence of it in diesis (sharps)? I couldn't find it myself. Is there a name for this phenomenon so I can learn more?

There isn’t a name for this phenomen. But we can find one. In German I would call them “gleichnamige” Tonarten, in English this would be “same named” keys.
(Now, as we know they are not exactly the same name, as the “related” key*1) has added a -bemol (flat) and is a half tone lower!)

The phenomen can be explained quite simply by the circle of fifths and by the tones of the twelve
tone scale (fixed do names!)

Do, Re-bemol, Re, Mi-bemol, Mi, Fa, Fa#/Sol-bemol, Sol, La-bemol, La, Si-bemol, Si.

Of each tone of the doremi scale exist two keys: one on the -bemol site (flats) and the -diesis (sharps) site. The two keys with the same name are differing logically a minor second respectively 7 fifths
and can be played by exchanging the amount of # with the amount of b of its same named (“related”) key:

So this “related” keys in question are:

Si-bemol (2bemol) and Si (5#)

analogically we get (starting with Re-bemol):

Re-bemol and Re

La-bemol and La

Mi-bemol and Mi

Si-bemol and Si

Fa and Fa#

Do and Do# (Re-bemol)

Sol and Sol# (La-bemol)