Just using patronymics was problematic for various reasons, taxes, record keeping, etc. So, many countries standardized on some kind of surname over time.
Denmark, for example, mandated surnames in 1828. Having done a fair amount of Denmark geneology, all those Pedersons and Pederdatters can become confusing.
I think the only way to have a situation where we DON'T use surnames would be:
1) Just have much smaller, less bureaucratic societies such that fewer of the confusion issues arose
2) Develop some kind of standardized patronymic-style system that avoids some of the confusion. For example, you could have name, patronymic, and then a further qualifier (city or profession), such as Matthys Jansen Van Keulen, translated, means: Matthys, Jan’s son, of Cologne. Or, Petrus Jakobsen the blacksmith could be Petrus Jakobsen de Smit. Of course, that's where a large percentage of surnames came from.
So, hard to imagine a non-surname type system in the modern day, but you certainly have some societies/languages that still include the patronymic as the middle name, e.g. Alexandr Dmitrovich Borodin.
Its not harder to tax Bob son of John (or Bob Johnson as the case might be), than it is to tax one of many Jack Smith's or equally many David Gracia's, so while at the face of it that arguement makes sense, it fails at closer examination, given that we have so many with similar names, that going solely by name is a folly.