Let me preface this by saying that I am not a professional historian; however, I am an experienced home brewer with an enthusiasm for history. So I can at least offer that perspective with some references. I also live in Germany, for what it's worth.

I think a few terms will be helpful to define before addressing your question specifically. Beer is first made by steeping crushed, malted barley grains (or some times other grains such as wheat) in hot (but not boiling) water. This process allows the conversion of the starch in the grains to simple sugars. At this point in the process, the liquid is very sweet! After this step, the grains are removed from the water/sugar mixture (called the 'wort'), then the wort is boiled (note: strictly maintaining a boil may be a relatively recent practice, but this isn't hugely relevant to answer your question). During the boil/heating, a variety of botanicals can be added to the wort. The purpose of this is to balance the sweetness of the beer with the bitterness of the plant.

To me, this last part is the key to your question. Historically, a variety of herbs, fruits, and/or spices were used to fill the role of bittering/flavoring. In modern beer, hops are almost universally used. I would argue that the use of hops is the defining factor that would make a fermented malt beverage identifiable to a modern drinker as "beer." Records of hop cultivation go back to at least the 8th century. However, it's not until around the 15th-16th century that that we see widespread and consistent adoption of hops in beer brewing in Europe (although hops were adopted a century earlier in Belgium and the Netherlands). In fact, in 15th century England, beer and ale were two distinct terms, with beer referring to a hopped beverage and ale unhopped. This distinction no longer exists; today, "ale" simply refers to any beer made with ale yeast (as opposed to lager yeast, which ferment at lower temperatures).

Focusing again on Germany, the famous Reinheitsgebot (the purity law) was established in Bavaria in 1516 (although there were some earlier laws that were similar). Among other things, this stipulated that beer could only be made with hops, barley, and water (and later yeast once microorganisms were discovered; restrictions on other grains such as wheat were also loosened). One of the more interesting factors influencing the use of hops was religion. In Germany, there was no tax on hops by the Catholic church, and it has also been suggested that the promotion of hops was intended to curtail the use of herbs viewed as playing a role in pagan rituals.

Certainly, you can still find fermented malt beverages today that are flavored with herbs other than hops; these are collectively referred to as gruit, although the term itself is really the name of the herb mixed used in the brewing process. Additionally, there are many modern and traditional beers (i.e., hopped fermented malt beverages) that do not follow the Reinheitsgebot; for example, some Belgian beers use coriander seeds and sugar in the brewing process. However, you asked about Germany, which today is dominated by light-colored, lower-alcohol lagers such as Pils and Helles, beers that have been copied many times the world over. I would speculate that such a beverage first comes to mind when most people think of "beer."

In any case, although one could certainly find hopped beer before the 16th century in Germany, it would have been much easier to find a clearly identifiable modern beer after this time, especially given the co-occurring transition towards more mass production of beer, which would have ensured more consistency among batches. Indeed, many of Germany's most famous breweries emerged from Bavaria during the 14-17th centuries, producing the pale lagers and wheat beers that are familiar to modern drinkers.

A final edit for clarity: As /u/Daedalus0815 rightly pointed out in a follow-up below, Germany of course did not yet exist as a unified country in the 16th century. Thus, if you really want to know about the beer you'd find in Berlin per se, the developments in Bavaria at this time would not necessarily apply wholesale. Northern German and Southern German (e.g., in Bavaria) beer brewing differed at this time, and continued to do so long after. Today, German brewing practices largely center around lager beer (made with yeast that ferments at a cool temperature) and wheat beers (and increasingly "craft beer" style breweries are popping up). However, there are still popular beers today that fall outside of these categories, such as Kölsch, which is a lager-like ale from Cologne. Anyway, in 16th century Berlin, you would have found much more variety in brewing practices than you would have in Bavaria. However, much of this detail might fall outside the scope of this answer. I think it depends on whether you mean, when would it first have been relatively common to find a recognizably modern beer in Berlin, or when would virtually all beer available in Berlin have been recognizably modern? I find the widespread changes occurring in brewing in the 15th-16th century (not just in Germany, but in, e.g., England, as well) to be at least a good starting point as an answer to your question.

References:

Corran, HS. (1975). History of Brewing. David and Charles PLC.

Oliver, Garrett. (2011) The Oxford Companion to Beer. Oxford University Press.

Unger, Richard (2004). Beer in the Middles Ages and Renaissance. University of Pennsylvania Press.

Edit: Reworded a few things for clarity.

I am also not a historian but am an avid homebrewer, and have read up on the history of brewing quite a lot.

The adoption of hops over herbs to bitter beer has already been mentioned. Additionally, hops started off in Germany and Bohemia, later migrating to England and elsewhere, some time in the 1500s. So yeah, hops have been a part of beer in Germany for a very long time.

Another, arguably even larger impact on the taste of beer is the advent of coke (the fuel, not the drink). Grain like barley is soaked in water, allowing it to partially sprout, and then kilned to dry it out and preserve it. This process is called malting. When the grain sprouts, amylase enzymes are created, which eventually allow the starch to convert into sugar. We're more interested in the drying phase though, as this is what creates the flavors you associate with different styles of beer. A dark roast gives you the flavors needed for stout, a light roast gives you the flavors needed for lagers, etc.

Prior to the adoption of coke (a refined, smokeless version of coal), maltsters used wood or preferably straw to kiln their malts, which added a smokey, earthy flavor. If I have time later today, I will dig through a few books I have with quotes and comments from the day on the smokey flavors of beers. Beer was often dark in color, and a rough drink.

Coke is said to have been invented in China in the 4th century, but in Europe it was invented in Derby, England in the 1600s. It quickly became popular for kilning beer as the smokeless heat it gave off allowed for a more pleasant flavor in the beer. These malts became known as "pale malts" and were used in "pale ales."

Around 1640 is when coke took off in England, and I found a fun quote from 1637, a petition to Charles I to test out the use of smokeless coal for kilning malt

61. Petition of John Gaspar Wolffen, his Majesty’s servant, to the King. Your Majesty gave leave to petitioner to make trial of his invention for brewing with a “charked” sea coal, which, as your Majesty has seen yourself, yields no smoke, and will do as readily, and within a little as cheap, as the ordinary way of brewing. Prays licence for brewers of Westminster and other places, questioned about smoke, who are willing to embrace the said invention, to continue in their brewhouses without molestation.

I'm not sure how long it took for coked malts to make their way to Germany, but I can tell you that over 100 years later, people in England were still commenting on the fine quality of coked malt over straw kilned malt, so it may have taken some time to spread.

It is a most sweet Fuel for drying Malt, the pale Sort in particular, but is best made from the large Pit Coal, which has supplanted the Use of Straw Fuel; and, when it is made to Perfection, it is the most admired Sort of all others

from the London and Country Brewer, 1759

By 1842 the Pilsner had been created in Pilsn, and the German Helles lager was created shortly after, so it's safe to assume coke was in widespread use by that time.

So to answer your question in short, prior to the mid 1800s you probably wouldn't recognize German beer because you wouldn't find any Pilsner or Helles lagers. Prior to the mid 1650s you definitely wouldn't recognize German beer because everything would have a slight smokiness to it if kilned with straw, or an intense smokiness if kilned with wood. It was a great time for Rauchbiers though.