Why, in general, do product or service prices rise in time? My question is simple:
Why is it, in general, that the prices of products or services (regardless of demand and supply) are always on the rise in western society? For example, the rent we pay increases every year. The ticket prices for public transport are getting higher every year. The costs of living: higher every year. On the other hand, money income from labor also rises. Why can't they stay stable? Is this inherent in the western economical model?
Or is it because of the constant battle between employees and employers? Which leads to a spiral of ever-increasing prices. Employers ask higher prices, employees higher wages, after which the employers increase the prices, etc.
Government policies can intervene, of which communism is the ultimate example, but that's based on a non-capitalistic economic model, though in the former USSR the number of products increased just as well, although they were state possessed. The visible material world in the former USSR, for example, how the cities and infrastructure looked like didn't differ, that much from how our cities look like. Economic growth and activity took place there too. Hospitals improved, technology in general (the space program among it) made progress, there was a separation between people who lived in bigger state-possessed houses (these were the people who had the power in the state: the employers) and cars different from the Lada, obliged to buy for the employees in the state. It was more uniform though and those who diverted from the norm were put away (as is the case in western societies, calling themselves somewhat arrogant, part of the first world, and communist countries the second one; in current times the borderline between the two has blurred or has vanished altogether). The system, although totally state-controlled, did provide for everyone (as long as you conformed, but that's the case in western societies too, where in fact it provides for everyone too, and also with huge differences between the employees or the employers). And it's a fact that the gap between the people who have a huge amount of private possession and those with a small amount is ever-increasing. Which was also the case in the former USSR but not on such a big scale as in the western world in which we now live. And who cares if you live in a privately owned house or a state possessed house? Only the rich people here, I guess. The more they possess, in general, the more they long to possess, like an addiction. Let it be clear though that communism, as well as capitalism (just look at the national debt of the U.S.A.: 14 000 000 000 000$!), is not my cup of tea!
All of this as an aside though.
Prices increase over time generally because of inflation. Money does not have constant value over time $\\\$1$ in 2010 is not equal to $\\\$1$ in 2020. As you mentioned in your question you can observe that not only prices of products and services increase but also wages. This is precisely because money is becoming less valuable so the same value embedded in work as well as in products and services require more money.
There are various reasons why value of money might decrease over time. Following Blanchard et al. Macroeconomics an European Perspective (a standard undergraduate macro textbook) the money market equilibrium (where the value of money is determined) can be expressed with the equation of exchange as:
$$MV=PY$$
where $M$ is the money supply, $V$ velocity of money, $P$  the overal level of prices  in the economy and $Y$ real output. This model is a bit of an oversimplification in more complex versions expectations of these variables are important as well but for this question it will do. We can solve the model for prices to get:
$$P= MV/Y$$
Hence prices can change either if:

$Y$ drops. This is because if the number of goods and services in the economy decreases but the amount of money and velocity remains fixed the same amount of money will chase lower amount of goods and services which will bid up all prices across an economy. However, it is worth noting that this is not the cause of inflation that we see over long period of time as most economies are on net basis growing (i.e. $Y$ is increasing) with an occasional recession which interrupts the growth.
$M$ increases. The mechanism is similar as with drop in $Y$ but now the amount of money increases in the economy and the number of goods and velocity remains constant. In modern economies $M$ is controlled by government institutions most often by central banks which can contract or expand $M$ directly or indirectly in various ways. Over long periods of time most economists believe that inflation is caused by increase in the money supply so this is the major reason why you will see the inflation going up over long periods of time.
If $V$ increases. Velocity of money is the average number of times a bill is used to purchase goods and services. It tells us how fast are money circulating in the economy. Generally the more faster the money circulate in the economy the higher will be the inflation rate. $V$ changes over business cycle, generally dropping in recessions and increasing in expansion but over long periods of time it remains more or less constant.
Some combination of all of the above.


Why can't they stay stable?

They can be stable. This is ultimately a policy choice, there were historical periods of near zero inflation and in principle central banks/governments, which control $M$ around the world could achieve long term price stability if they wanted (even though they might not be able to do it precisely every single year due to random economic shocks they can do that on average over business cycle).
However, as pointed in the link Brian provided in his +1 comment most economists believe having certain level of inflation is desirable. This was already answered excellently by FooBar so I wont be repeating it here.

Is this inherent in the western economical model?

First of all what is 'western economical model'? Currently almost all countries around the world have a mixed economy - a system that combines free market with central planning. For example USA, South Korea, Japan, UK, Denmark, South African Republic, Russia or China are all mixed economies. Countries differ in the quality of their institutions in policies they follow and in many aspects but looking at the big picture most countries around the world follow the same economic system. Few exceptions would include countries such as North Korea.
Second, inflation is actually  much lower in 'the west' (i.e. Europe and North America and other countries that are colloquially known as 'western' such as Australia or New Zeland) then elsewhere as the data from IMF for year 2019 below show. Feel free to look at any recent year in the IMF dataset and 'western' countries will be always among those with lowest inflation rate.
This is however not due to following some fundamentally different economic system but rather due to different institutions. Most western central banks are independent from government with a mandate that either completely or partially requires them to maintain low inflation. Also just to be clear I am guessing that by 'west' you are not meaning just a western hemisphere which includes many African or South American Countries, but rather you mean nations with roots in the Western Europe (i.e. USA, UK, France, Canada, Australia etc.).
    Region                       Average Inflation Rate in % 2019
    South America                      no data
    Western Europe                      1.3
    Australia and New Zealand           1.6
    Europe                              2
    North America                       2
    Central America                     2.2
    East Asia                           2.3
    Southeast Asia                      2.3
    Caribbean                           2.8
    Pacific Islands                     3.2
    Asia and Pacific                    3.3
    Western Hemisphere (Region)         3.4
    Eastern Europe                      3.9
    South Asia                          4.8
    Middle East (Region)                7.9
    North Africa                        8.2
    Africa (Region)                     9.2
    Sub-Saharan Africa (Region)         9.7
    Central Asia and the Caucasus      11.9