What is the oldest known recipe still being brewed?
No, there's a profound difference in modern beers, which for several hundred years have been more or less dominated by the flavour of hops, which contain bitter acids that create much of the taste and aroma you probably associate with beer. Earlier medieval beers were often flavoured with blends of herbs known as gruit; these are available as craft beers, but for most people more as a curiosity than as any significant portion of the market. There are also occasional tiny batches of very ancient beers, or they can be reproduced by reasonably experienced homebrewers if you have access to the equipment. As that link discusses, we do know a decent amount about very old recipes, but chances are good that they will prove unpalatable or highly odd to a modern consumer.
Also worth considering are that there have been significant changes in the flavour and chemical profile of the grains being malted, some (such as emmer in the story I linked) are very difficult to acquire malted, and carbonation is significantly easier to trap with relatively modern drafting and bottling technology, so the carbonated sparkle, a necessary product of fermentation, would generally have been far less present in early beers.
This question really touches on what I've been spending my last few years studying. My actual research is a few hundred years later and focused in Scotland but the lack of resources for there means I can cover this.
The short answer is no. I don't know how much you personally know about beer and modern brewing so you'll have to excuse my covering what you may already now. By craft beer rules, beer is divided into ale and lager depending on if the yeast sinks or floats during fermentation. Most of the really large and popular brands are lagers, both meaning that the yeast sinks during fermentation and that it requires cooler fermentation temperatures. Lager did not really get created until the 15th and 16th century so it's already different from most modern beers due to that.
And that's when you get into ales, or just top fermenting yeasts in general. Ale in 1016 wasn’t ale today. Historically, the ale and beer divide comes down along whether or not hops are used and actually have different terms (cerusia without, and bier with). It actually means that most beer consumed at the time was more properly referred to as ale. Beer really didn’t start to become a major competition for ale until a maybe 800 or even 600 years ago, and exactly when depends strongly on region. Richard Unger (in his book Beer in the Middle Ages and the Renaissance) mentions a few locations started to know about hops and their use in brewing in the 9th century. One example he gives is that the Carolingian abbot Adalhard of Corbie discussed the use of hops in beer in 822. It’s possible you could find beer then but a few examples don’t really mean it’s common. I happen to know Scotland didn't generally produce hopped beer until the 14th century and even that was mostly imported beer and beer made by foreigners. Admittedly Scotland took to hops later than a lot of places but 1016 was too early for it to be common. Instead it’s far more common for ale to be flavoured with something else or simply be made of water, malt, and yeast. Gruit is often mentioned but it’s a specific herbal combination used only in a few places and still arguably new.
Even once you’ve established that it was hopless, it’s only the start. The brewing technique was also fundamentally different and many of the ingredients were also not as they are now. I’m minimising discussions on the modern techniques on brewing as much as I can but I think it’s important to have context to help make the differences more clear.
Firstly, technique. Step 1: Making a mash. There’s no temperature stability and without a strong control over temperature during mash you’re not able to really get a stable wort. The earliest actual written recipes I’ve read (18th century) have suggested controlling temperature by getting the water to a boil, putting a lid on it, and checking it occasionally by lifting the lid and seeing if you can see the bottom of the pot. It would lead to differences between batches. After that stage currently you'd sparge (pour hot water over the grain after the mash) to make sure you'd get all of the sugars off the malt but the hypothetical brewer in 1016 would instead make several batches with one batch of grains. The first batch would be strongest and each one afterwards is weaker. When attempting to make a recipe by this method, Peter Brears (Cooking & Dining in Medieval England) ended up with three batches. The first batch ultimately fermented to 10%, the second 3%, and the third was under 1%.
And technique here also is different because of the lack of hops. Using hops requires a boil after the mash and strain and it has the side effect of helping clarify the beer. As they were not using hops, they didn’t do a boil and so the liquid they were fermenting was more cloudy. And where yeast came from was a mystery. Brears supports the idea that liquid yeast was skimmed off the top of previous active batches and used in the newer ones. It works but it more often leads to infected batches that either end up sour or flat. Also, without the preserving power of hops, fermentation time was shorter. Brears argues that the 1st batch of his could have been aged but there was a preference for younger ales. Whether it could last or not is a different question but the short fermentation means more cloudy as well. Without secondary carbonation it was also only marginally carbonated.
I could also get into the locality of production and who was making it (women and monks) but the last thing I’ll get into briefly is malt and how it means ale was likely kind of amber. While the malt was likely barley, and maybe oats, the technique for malting didn’t lead to an evenly dried and roasted product. The most basic malting chamber involved drying the grain on basically straw over a fire. It not only was at risk of catching fire but it didn’t allow for even colouring. Even a relatively small percentage of dark or even moderately roasted malt would disallow for pale and golden ale.
I’m sure I’ve managed to forget something but I hope that answers most of your question. Feel free to have follow-up questions or ask for clarifications.
No, in fact, even beers from less than 200 years ago will likely taste very different to modern beers.
Thermometers, hydrometers, single yeast cultures produced in labs, basic principles of hygiene and sanitation, automatic temperature controls, etc., things that even home brewers today take for granted, were not available to the average commercial brewery, in some cases like single yeast culture, until late 19th, early 20th century.
The use of thermometers and hydrometers, for example, didn't become standard until mid 19th century. When Czech brewer František Ondřej Poupě proposed hygiene and sanitation at the turn of the 18th and 19th centuries as practices that could improve the quality of beer, he wasn't taken seriously at first.
To this we should add that barley cultivars are different than they were centuries ago, and that malting has also evolved. The same could be said about hops and their processing. And we aren't even speaking about filtering and pasteurisation here.
The closest thing you could find today to what beer might have tasted like a few centuries ago are the Raw Ales that are still being brewed by some people in Scandinavia and the farmhouse ales in the Baltic Countries. In many cases, they are still using old technology and processes, and even heirloom yeast cultures. This blog speaks almost exclusively about them, and with a lot of solid research. (This might also answer the question of the oldest recipe still being brewed).
For something more "modern", Ron Pattinson is perhaps the best source. His blog is full of tables and transcriptions from primary sources (brewing logs, etc.), and he has helped several renowned breweries to recreate old recipes based on archives and logs. I think he has even managed to get someone to grow a strain of barley that nobody had grown for ages, and use it to make brown malt with the practices of the 18th century, or at least it was one of the projects. I recommend his book The Home Brewer's Guide to Vintage Beer with recipes of famous beers from the 19th and 20th. It's a fantastic way to see how beer styles have evolved in time.
You might enjoy browsing this Sunday AMA from a few years ago until someone comes here to more specifically address your question