I noticed that when I stretch my neck to the left side, I can feel the
stretch in my right-rhomboid (where the pain is). when I do push- or
pull-ups now, I don't feel the pain.
Are you sure it's your rhomboids? And not your traps or levator scapula or something else? You shouldn't feel much of a stretch on your rhomboids simply by pulling your neck to your opposite side. Rhomboid minor does attach to C7 but that's the start of your neck and you typically need to have (pro)traction on the shoulder blade to stretch the rhomboids. They are pretty deep between the shoulder blades and the spine, difficult to stretch.
So I've been doing push-ups for over six years and the first five
years it was quite simple 6x20 sets of push-ups, without any pull-ups.
Now, for the past half year I have increased sets and amount, and
added pull-ups to my training routine.
Without knowing much about your overall routine currently, if you're doing more than 6x20 now (and even 120 reps a day is a little excessive most of the time) it's possible you simply have too much reoccurring volume on the push ups. Regularly doing hundreds of reps of the same movement over and over again with little variety appears to increase the potential for overuse injury.
I usually do a warm-up jog of 5 to 8km, but yesterday, I finished only
4km and walked-out for the remaining 2km. Yet, I completed my full
push-up training. So I presume it's from the push-ups.
See my point above. You don't really have to know what exercise 'caused' it, it doesn't matter what caused it, you have to figure out why it's being cranky so you can fix it. A lot of those muscles are involved in both the push up and pull up anyway, so regardless of cause, both could continue to create pain for a variety of reasons.
But also, a warm up of 5-8 km running? Why? That's not a warm up, that's a training session in and of itself. A warm up for resistance training is like 5-15 minutes and I'd say you probably want something more specific to your training as a warm up, not just running.
Unless the training goal is specifically running and push ups/pull ups are an afterthought. In which case, the push ups are definitely excessive volume. That being said, fatigue always hurts performance. There is no way you're not a little tired after even a 4km run. So it could very easily also be a technique issue as the result of too much fatigue.
It's possible that either exercise could lead to such an injury or at least pain. Pain doesn't always correlate with an actual injury and actual injuries don't always yield pain. Pain is more of a warning system and it's prone to faulty wiring sometimes.
For the time being you should take some time off from anything that aggravates it and likely see a physical therapist to determine what's actually going on. They can help you formulate a plan of action for return to your routine.
Now that the rhomboid (assuming it's the rhomboid) is cranky both exercises will likely aggravate it anyway. It needs time to heal.
When it's healed then consider some of the points above. Your routine lacks movement variability, clear progressive overload (6x20 is pretty static), likely a good warm up, there may be some technique issues and more than likely you're simply doing too much inefficiently.