As the title implies, have (with any PoD) the medieval period in Europe be defined by republicanism, instead of Feudalistic Monarchy. What would be the social, economic, political and philosophical implications of this? What types of republics would medieval states adopt? A lot of early medieval European cultures had people's assemblies (the Germanic Thing, the Slavic Veche), so I can definitely see that developing into a robust Republican framework (like it did in Greece and Rome).
While most have concentrated on Roman heritage and thus Roman PoDs in reply to this message, and while I can see their point, I'll go ahead and suggest a much later divergence is possible.
There was a communalist movement sweeping Europe in the High Middle Ages, beginning in the early 12th century and never really subsiding. It achieved the establishment of town councils, mayoral elections etc. Thing is, if you want the Middle Ages as you know them, you'd have to face the option of this being a sort of "feudal" make-up, too, as it was IOTL: the Free Cities owed the Holy Roman Emperor contributions just like any imperial knight, count or duke did. Now IOTL, much of this happened within the HRE, where "city rights" and granting the status of free imperial cities was an outlet, especially in the collapsed Duchy of Swabia. The peak of success of this movement was the establisment of the Swiss Confederacy, which, for centuries, still considered itself and was considered a part of the HRE, albeit a special one. And the members of this confederacy acted as feudal states in their own rights, acquiring right-less "possessions" for example.
When the success of the Swiss model became evident and conflicts between slowly consolidating territorial principalities evolving towards modern statehood and communes pushing to preserve and expand their liberties and privilegies escalated anew in the late 14th and early 15th centuries, "Helvetisation" was a slogan of the day. Italy was full of republics anyway (including Adriatic Dalmatia). If some of these conflicts had gone differently (i.e. towards the "Swiss" version), then it's quite possible for the HRE to become primarily a super-confederacy of leagues of towns, where territorial principalities would continue to exist but not be "the norm". Imagine the seven electors joined by eight new ones: from the Hanseatic League, from the Swabian League, from the Swiss Confederacy etc. Add more regular Reichstage, and you're on the way to a collection of republics electing themselves a proto-constitutional monarch.
The breakthrough for the communal movement which IOTL did not come but which would have been necessary here would be its success in France, too. I know too little about Medieval France to say how this could plausbly come about, my impression is that a centralisation of the monarchy only evolved during the last decades of the 100 Years War, but long before that, say in the 12th and 13th centuries, the conditions for more successful communes in France are unclear to me.
The social implications of this would be propertied farmers, crafters and merchants seeing themselves and being seen as the pillars and foundation of statehood and power. Look at the various guild wars and internal conflicts in the Old Swiss Confederacy for guidance as to how conflictual their relations would still be. But the conflict lines would run differently from OTL. IOTL, capitalism developed - depending on your theory - either in spite of or due to centralising territorial statehood. I have come to tend towards the former theory (this has been a long theoretical march for me) although I acknowledge the massive boost of the colonisation of the New World by territorial monarchies. Well: Going by this road, and assuming the Americas are still discovered by a Spain and Portugal which are less affected by the changes outlined here, then I'd say we might see developed capitalist structures earlier on. The dividing line between those participating in power and those without power would be more clearly along property lines.
The political implications of this are diverse. On the one hand, you'd have a lot more open political debate than IOTL. On the other hand, it would be a lot more parochial and pragmatic. Modern political philosophy evolved in the context of and in the struggle against absolute monarchies. Instead of Locke, Hobbes, Rousseau etc., you'd have entirely different philosophers and philosophies - at some point in time, whether earlier or later depends really - emerging from and capturing the kind of struggles and challenges within and between city-republican states. I'd imagine less grand designs and abstractions, and more narrative integrations of achieved traditions and age-old conflicts and their resolutions.
As for philosophy, that is way over my head right now.
There would be a clear break between pre-medieval Things and these High and Late Medieval city states, as it was IOTL.
As for the Veches, well, for Eastern Europe you'll need an additional divergence probably. Mongol rule is a major factor here that cannot be glossed over.