My players want to be able to identify a spell being cast by another spellcaster like it was possible in 3.5 with the Spellcraft skill. So are there any rules, spells, feats, class features or anything else for that in 5th Edition?
Depending on what's going on, this is probably simply handled by an Arcana check or Arcana based challenge. It is not currently handled explicitly by rules or mechanics (no rules/feats/class features or anything else for it).
If your caster observes the spell being cast, and it's effects, they most likely already know the spell if it's on their class list, or can identify it with an arcana or religion check (depending on if it's on the wizard or other arcane caster's list or on the Cleric's list, nature might also be appropriate if a spell only shows up on the Druid/ranger list).
That said, if they observe the spell's components (material, somatic, verbal etc), and the effects are too subtle to notice, or they aren't immediately apparent for some other reason, then a check of the requisite type is definitely called for.
Again, I think if you only observe the effects, same story as above, you need to make a check or several.
Lastly, if you observe only a single component of a multi-component spell, I'd raise the DC or invoke disadvantage on the check(s).
Ultimately, this is the kind of thing that rests squarely on the shoulders of DM discretion. There isn't a mechanic that dictates how it works right now (maybe in the DMG?), but for right now I would handle it very similarly to how knowledge checks work, no need for additional actions or anything if you're actively watching the caster (say, like in combat). I would make learning the spell's name easy, the effects moderate and the level difficult.
Xanathar's Guide to Everything, released Nov 2017, contains a specific answer to this question.
TL;DR: Use a reaction as the spell is being cast to identify it, or use an action on your turn to identify a spell by effect after it was cast. Int (Arcana) check, DC 15 + spell level. Advantage if the spell is on your class list.
Identifying A Spell
Sometimes a character wants to identify a spell that someone else is casting or that was already cast. To do so, a character can use their reaction to identify a spell as it's being cast, or they can use an action on their turn to identify a spell by its effect after it is cast.
If the character perceived the casting, the spell's effect, or both, the character can make an Intelligence (Arcana) check with the reaction or action. The DC equals 15 + the spell's level. If the spell is cast as a class spell and the character is a member of that class, the check is made with advantage. For example, if the spellcaster casts a spell as a cleric, another cleric has advantage on the check to identify the spell. Some spells aren't associated with any class when they're cast, such as when a monster uses its Innate Spellcasting trait.
This Intelligence (Arcana) check represents the fact that identifying a spell requires a quick mind and familiarity with the theory and practice of casting. This is true even for a character whose spellcasting ability is Wisdom or Charisma. Being able to cast spells doesn't by itself make you adept at deducing exactly what others are doing when they cast their spells
-Xanathar's Guide to Everything (pg 85)
For the likely follow-up question of not having another reaction to cast Counterspell, Jeremy Crawford says it's intentional.