Blast furnaces have been in China since the 1st century, and arrived in the west during the late 1400's. Imagine now a society "stuck" in the early modern / late medieval period, could they produce steel alloys, particularly the "Mangalloy"?
Manganese was the first alloy steel
My research into steel-making methods of the time is somewhat inconclusive, but I am reasonably confident that the heating method that Hadfield used as the Siemens-Martin process, also called open hearth furnace.
Ancient smelting methods
First, a rundown of various ancient smelting methods.
Bloomery: The oldest type of furnace, this is just a pit or chimney made of clay or stone. You put the iron in, heat it with fuel and a bellows, and then take out the glowing mass to form into the shape you want. The furnace can't really be scaled up, and it won't fully melt the iron. This process converts ore directly into
Cupola Furnace: The Chinese never developed bloomeries, instead they went straight to casting iron. It is not completely clear what sort of forge they used, but evidently it could reach 1130 C and and melt iron ore. This can be poured into molds and cast.
Finery Forge: Cast iron is brittle and nearly useless, so within a couple centuries the Chinese invented the finery forge, where cast iron is heated and repeatedly beaten to remove impurities to make wrought iron.
Blast Furnace: In a blast furnace, fuel is constantly fed along with forced air to run off slag and form pig iron directly (which would then be finery forged into wrought iron). This greatly increases the rate and fuel efficiency of iron production, and obsoletes the simple cupola.
Puddling Furnace: Instead of pounding with a finery forge, the molten iron is melted without contact with fuel and then stirred up with iron rods (which then melt into the mix). It can then be poured and cast into wrought iron. The key is the without fuel part, this allows free carbon to be burned out with oxygen, and so wrought iron to be made. The Blast Furnace-Puddling combo was in use in China by around the 1st century AD, and not copied in Europe until much later.
All of these techniques could be used to make steel. Any resulting wrought iron could be hammered (similar to the bloomery or finery forge) to decarbonize until steel was made. This was normally done under a 'cold blast' or forced air.
The Europeans really didn't catch up with the most advanced Medieval Chinese blast furnace until the combination of Abraham Darby's coked blast furnace in 1709 and Henry Colt's puddling furnace in 1784. After that advances came rapidly.
Hot blast: Invented by James Neilson in 1828. This is a pre-heating of air before entering the blast furnace with a regenerative heat exchanger. This is key for both increasing fuel efficiency and increasing furnace temperatures. In this old photo, air is run through three Cowper stoves which capture the exhaust from the furnace and use it to heat incoming air.
Bessemer Process: Invented by Henry Bessemer in 1856, this made use of the hot blast to blow hot air through the molten iron. This completed the puddling process at the same time as the iron was being melted, and control of air flow allowed the carbon content for a desired steel product to be reached at the same time. This combined all three steps of the old Chinese process into one (blast furnace -> puddling -> cold blast forge) to make steel much cheaper.
Open Hearth furnace: Invented by Carl Siemens and Pierre-Emile Martin in 1865. This makes more extensive use of regenerative pre-heating to achieve fuel savings, and to increase temperatures even further, enough to melt steel. This allowed it to be used to recycle scrap steel, but the slower heating process also allowed time for extensive sampling of the product to achieve the desired chemical compositions
I suspect the open hearth furnace is required to make a manganese alloy. The only way to ensure you have the correct amount of alloy mixed in, is to add it to molten steel, and you can't do that without an open hearth furnace.
You could add chunks of manganese to your ore and do the ancient Chinese process, but you would run into problems. As the iron mixes with carbon (from charcoal or coke) its melting point drops, allowing it to melt in a blast furnace. But as it mixes with the manganese, its melting point will raise again. Before hot blast, there was no way to keep this mixture molten. Thus, you would end up with an incompletely mixed solid. I suppose this could be pounded into an alloy blade, but that would take months of hammering per blade, and the alloy mix would be pretty uncertain. So only the finest (modern) Damascus steel-type blades could be made with an alloy, but it might have a bad alloy mix and end up weak or brittle.
So the conclusion is, you need the hot blast and open hearth technologies to have any hope at making Mangalloy. On the plus side, the technological requirements of a regenerative heater for hot blast are not much. You can make a brick pre-heating chamber and use a waterwheel to force air through pottery inlet air pipes through it to get a pre-heat.
So final conclusion is: Yes, you can make alloys if you take Medieval Chinese steelmaking technology and give them a method of regenerative heating of blast furnace inlet air.
Unlikely but possible
Although the Chinese may have had the base level technology I doubt that this technology was very “robust”, meaning the output would have been a bit variable depending on who was making it and what went into the furnace. I also expect it would have been expensive and labour intensive so I doubt they would have had the luxury of being able to play around experimenting with a vast array of steel making materials.
It is just possible that it might have been different, but they would have needed a lot of luck. Assuming that they did have some time to experiment, perhaps some high ranking nobleman had an interest and allowed his workers to experiment.
Then assume they had the right materials to hand to experiment with – with high manganese content ore uncontaminated by phosphorus, arsenic, or sulphur.
Then assume they put a lot in, not 4-5% leading to very poor quality steel, that will just shatter if hit with a hammer, but around 13% to make Magalloy.
To add to the other answers, the Japanese were great at making hardened steel.
At the bottom of that page in the "Notable swordsmiths" section, the first listing is for Amakuni, who is reported to make the first single edged sword in 700CE.
Japanese swords were/are made not just by pour metals, but by pounding and folding the metals, effectively mixing the metal in a solid form. Folding the metal means that anything not mixed while liquid effectively gets mixed in anyway. Folding means that the layers go up by powers of 2. So, fold a piece of paper in half 2 times gets you 4 layers, 4 times gets you 16, and folding 16 times gets you 65,536 layers.
Also, they would start by forge weld different qualities of steel together to get the right hardness for their purpose.
These swords were made over a period of weeks, not the months suggested by @kingledion to mix solid metals.
I think the answer is still yes, and a lot easier and faster than previously suggested.