The cambion's spear attack does fire damage, but no rule says that the spear itself retains that property.
A statblock only defines what a creature's attacks do, not the game statistics of any items they may carry. In this case, the cambion may make a spear attack dealing 1d6+4 damage plus 1d6 fire damage, but there is no rule or descriptive text which would define that the spear retains that property.
The ethos of D&D 5th edition is that a rule says what it says and no more.1 Therefore, the rules say only that the cambion deals extra fire damage with his spear attack, not that the spear itself deals that damage in anyone else's hands.
It's also not common (perhaps unprecedented)2 for a creatures in D&D 5th edition to have a specific magic item in their statblock. Treasure from encounters is handled under its own rules in the Dungeon Master's Guide.
Lore-wise, the cambion has many supernatural abilities, specifically including the ability to create and throw fire, so it's very easy to imagine that fire surrounds any weapon he wields; conversely, flaming weapons are quite rare and difficult to acquire3, so many would not have this specific item.
Of course, the DM's word is law, so you could as DM say that cambions do indeed carry actual magic flaming spears, but only if you want your player characters running around with a load of flaming spears after they defeat an army of cambions. For that reason, I don't recommend it.
1 This is a design principle put forward by D&D 5e core designer Jeremy Crawford, and is widely used as a guideline for DMs adjudicating ambiguous rules. For example, in How much fire damage does igniting Grease deal?, Crawford's interpretation is that Grease isn't flammable because the rules do not say that it is. This attitude contrasts with D&D 3.5 where the DM was specifically advised to interpret ambigious rules by inference from other similar rules.
2 The only instance I'm aware of where a monster is described as having a magic item is the flind in Volo's Guide to Monsters, whose descriptive text says "It wields a flail imbued with powerful magic by Yeenoghu himself". However, even in this case, the statblock does not describe the weapon itself, only the attacks which the flind can make. It does not specify what happens if a PC tries to wield the flail, nor even is it made explicit whether the flail is a magic item, or merely temporarily imbued as a warlock may imbue his pact blade. If a DM ruled that a PC gains all the flail attacks the flind had, that PC now has a weapon dealing 4d10 bonus psychic damage, which is exceptionally powerful for an item you might gain around level 9, and I do not believe this was the designer's intent.
3 According to the item guidelines, magic items are only found in hoards, rather than standard per-monster treasure. The guidelines suggest that the PCs will in total acquire around 18 rolls on the CR 5–10 table, or 3 hoards per level, and 80% of these hoards will not even reach level Table F, the first table to give a +1 weapon. A cambion could have a magic weapon, but most won't.