Lot of differing questions that one can go deep into, I'll try to answer best I can.

To what extent were these tribes the modern people of Europe?

Mostly for many non-Slavic groups (their predecessors migrated in a bit later), yes, they can be considered precursors in a rough sense, if we're going to be generous. Latin influences in the former Roman territories and from the Church also had an influence.

Also, how different were they?

The Alemanni, Suebi, Franks, Saxons, Lombards, and Burgundians were all, er, germanic (not to sound like a nazi). The Ostrogoths and Visigoths "emerged from the Gothic tribes, most likely a derivative name for the Gutones, a people believed to have their origins in Scandinavia and who migrated southeastwards into eastern Europe". They also could be considered roughly Scandinavian. (Fun fact, Beowulf would have lived in the area the Gutones came from)

Further migrators would be the early Slavs, the Bulgars, and the "Pannonian Avars" (no one really knows where the latter came from)

From the very little history I know of the time period, the barbarians all came in a similar timespan, from… somewhere. Actually, where did they come from? They weren’t natives right, that was the celts and later the Romans. So did they just appear?

They all roughly migrated from their homes outside the Empire's territory, in a period of time known as the Migration Period, mainly during the First Wave. Relevant Wiki Article. The main cause(s) of the Migration are generally regarded to be the invasion of Attila the Hun in 375, with the later ongoing destabilization of the Western Roman Empire later prompting further migrations. Later on, climate shifts, which had started in 150 AD and would intensify with the Late Antique Little Ice Age

Also unrelated to that, were the barbarians the start of feudal society, or did they adopt it?

Well that's a whole post on it's own, but to put it shortly, yeah, they sort of went along with the material trends leading to feudalism. The foundations of feudalism actually started with the destabilization of Rome; as Roman central authority waned, large landowners often decided that they would both stop sending taxes and hire troops to protect their own estates. This is often called 'manoralism' We can also see here it's brief history of its relation to feudalism:

Antecedents of the system of manoralism can be traced to the rural economy of the later Roman Empire (Dominate). With a declining birthrate and population, labour was the key factor of production. Successive administrations tried to stabilise the imperial economy by freezing the social structure into place: sons were to succeed their fathers in their trade, councilors were forbidden to resign, and coloni, the cultivators of land, were not to move from the land they were attached to. The workers of the land were on their way to becoming serfs.

Several factors conspired to merge the status of former slaves and former free farmers into a dependent class of such coloni: it was possible to be described as servus et colonus, "both slave and colonus". The Laws of Constantine I around 325 both reinforced the semi-servile status of the coloni and limited their rights to sue in the courts; the Codex Theodosianus promulgated under Theodosius II extended these restrictions. The legal status of adscripti, "bound to the soil", contrasted with barbarian foederati, who were permitted to settle within the imperial boundaries, remaining subject to their own traditional law.

As the Germanic kingdoms succeeded Roman authority in the West in the fifth century, Roman landlords were often simply replaced by Germanic ones, with little change to the underlying situation or displacement of populations.

The process of rural self-sufficiency was given an abrupt boost in the eighth century, when normal trade in the Mediterranean Sea was disrupted. The thesis put forward by Henri Pirenne supposes that the Arab conquests forced the medieval economy into even greater ruralisation and gave rise to the classic feudal pattern of varying degrees of servile peasantry underpinning a hierarchy of localised power centers.

So, yeah

Or did it just develop similarly across Eurasia?

Well, keep in mind that "feudalism" is a massive oversimplification of often very complex systems of labor relationships and obligations that had a lot of different names depending on the region of Eurasia, and these systems more often that not developed independently of each other. But yeah, there were some similarities in its creation.

Hope this answers your questions!