Is it possible to run out of Musical note combinations to make songs?

It's already happened - at least within certain parameters. Damien Riehl and Noah Rubin "brute forced" every possible melody using 12 quarter notes/crotchets on the white notes within a single octave (plus an additional tonic at the top of the octave to give 8 notes), generated midi files for it and saved them. They can now - theoretically - sue *ANYONE* because they made these 68.7 billion melodies. It would be up to the court to decide if the simple arrangement of "letters" is more important than the actual rhythm, many copyright lawyers argue that rhythmic exactness is not as important as the pitches used.

If those two guys can do that in only 6 days using one server it absolutely would be possible for someone to "copyright" something using all 88 notes on the piano, and most common rhythms for a short length of melody. It would take a lot of computation power, but it could definitely be done.

https://www.youtube.com/watch?v=sfXn_ecH5Rw

You cannot judge the infinity of possibilities in music, if you use commercial music as your standard.  YES - crap on the radio, or streamed to the mass market of sheep, will sound the same in each genre.  Many purveyors in each genre are indistinguishable - if you want to SELL, you need to go with what is popular among the masses - while "saying" you are different and creative.  Even orchestras concerts will always include some old warhorse works, so they can sell tickets, CDs, etc.  In the live performances that I and my colleagues do, we have to balance lesser heard, more challenging works, with things that are less confrontational or demanding in the listeners.  So we always have creative and startling things in rehearsal - but can't *fill* a program that way, because it alienates some people; neither can you fill a program with powder-puff works (for the listeners and yourself!) and keep integrity.  So true artists in ANY genre will never cease to be creative - people looking to make a buck will "stay in their own lane".  

Considering how many millions of songs that have been written, I suspect that it is becoming increasingly difficult to write an 8-note melody that is totally original, but when you add rhythm and harmony, the possibilities become significantly greater.  Hundreds of thousands of songs have probably been forgotten or very seldom heard.

I doubt it. Especially the cacophony I have heard claiming to be classical.

https://www.youtube.com/watch?v=3XfR6jQvvs4

I suspect you mainly listen to Western music which is based on the major scale and traditional timing schemes such as 4/4,3/4 or 6/8.  Other cultures have a huge variety of music which differs significantly from what you are used to hearing.  Try listening to Indian Sitar music.  I suspect as cultures merge and become more global, future music will incorporate these other cultures and change significantly from our traditional music.

I read that Sony Music has a music catalog containing over one million songs (!)

No, there are plenty of possible combinations, infinite really.  There is a lot of interesting and original music out there but we only ever get to hear a tiny percentage of it.  We only get to hear the ones that sound as similar as possible to the last one that sold well.  The music business doesn't want creativity, they want conformity so that they can sell the same thing over and over and over.  Music isn't dying off, it just looks that way because of the music industry's refusal to respect original or unique music.