As I read your question, you actually ask two distinct question, one whether house prices can be assumed to have positive trend, and one about whether population or something else causes prices to rise.
Is it simply a mistake to assume that house prices tend to rise?
Yes, this would be a mistake. There is no inherent economic reason why real house prices, ceteris paribus (i.e. all else equal), should increase or have increasing trend.
This does not hold just for housing but virtually any good, service or commodity. Prices are ultimately determined by supply and demand, and in turn supply and demand are determined by various factors such as peoples tastes, incomes, expectations, availability of substitutes (i.e. examples of things affecting demand), or factors like cost of materials, labor or other inputs and again expectations (i.e. examples of things that affect supply). If you are unfamiliar with how supply and demand work, you can also consider having look at first few chapters of Mankiw's Principles of Economics.
In principle there will always be some combination of the supply and demand that can result in housing price increase or decrease. For example, if tomorrow suddenly co-living with roommates would become hip and desirable, more people would live together, demand for houses would be lower and ceteris paribus price as well. Or alternatively, some new technology could allow houses to be built more cheaply it would expand supply and thus reduce price, ceteris paribus, as well.
Consequently, it would be grave mistake to just assume real housing prices can just go up. There is no economic rationale for such assertion. You have to first examine economic fundamentals and trends in economic fundamentals (like the costs etc mentioned above), and based on that you can make some forecast (but forecasts get progressively less accurate the more into future you want to forecast).
Is it tied to increasing population?
This is one of the factors, but not the only one or even a necessarily the most important factor.
Increase in population increases demand for housing, so this definitely pushes the real price of housing up, but demand is always just a half of a story, there is a whole cornucopia of examples where increases in demand were offset by even larger increases in supply leading to lower prices. In this case a lot of the increase in real prices of houses can be explained by supply factors.
Construction is one of the most labor intensive industrial jobs (Yang et al 2019). As real wages of workers increase it simply becomes more expensive for firm to build houses so they shift their supply to the left leading to higher prices.
Rigorous research aside, even if you just look at the cost of materials typical house takes to build they are not that expensive, but the labor that goes into them is. As real wages typically tend to rise over long-run (see FRED data here) you would expect prices of labor intensive goods to also increase in real term (all else equal). Of course, it is always possible that there is some sort of robot or 3D printing revolution just around the corner, and if that would happen the real price of housing would fall.
Part of the high price of housing is caused by bad government policies which restrict supply of housing. Many countries have on their books zoning laws or other policies that restrict the supply of housing. Such laws prevent firms from increasing supply of housing even if they would like to and thus lead to higher prices (see for example The Impact of Zoning on Housing Affordability Edward L. Glaeser, Joseph Gyourko).
Zoning laws are typically popular with voters because people who are already established in some area and own home have incentive to keep new house production down to protect their own property values. Also, some restrictions are made for environmental concerns (or at least under the pretense of such concerns like green belts in the UK). However, again as in the previous case, these laws might be changed in future, they are ultimately not very sustainable as new generations of voters will find harder and harder to become house owners and thus voter base for such policies will shrink.
Houses improve in quality over time. This can lead to sustained increase in real price, and even though typically price indexes are constructed in a way that at least a little bit tries to correct for this. Indeed if you click through the data sources you will learn the Fred data come from BIS which uses the same methodology as eurostat here, which does quality adjustments. Nonetheless, it is generally agreed that price indexes typically overstate rise in prices at least a bit (e.g. see examples of literature on this issue Baker Get Prices Right or discussion in Hwang et al 2004). Also, house prices are only recorded for sold houses but the houses people sell are not random. You will see new dwellings or newly renovated dwellings overrepresented, which skews the statistics.
However, the ultimate consequence of upward-trending real house prices would seem to be that, eventually, nobody on Earth could afford to buy a house.
No, this is incorrect assessment for several reasons.
First, even if there would be deterministic upward trend in house prices that would continue forever without any end the statement would not hold as long as real wages would increase faster or at the same pace as house prices. So the statement is logically flawed, rising housing prices do not necessarily mean people would not be able to afford houses at some point.
Second, economics is an endogenous system. The higher price of housing the higher pressure for people to change their taste (learn to cope with roommates, preferences for tiny houses etc), and more pressure for firms to come with innovation to be the first one bring cheaper houses and beat the competition, and at the same time the more pressure on bad policies to be either abolished or replaced by policies that promote house construction.
Historically people often worried that some commodity, like food, water or oil etc will eventually become so expensive nobody will be able to afford it and so far that fear is mostly unfounded thanks to innovation and productivity growth over time.