Several wands and other magic items in the DMG cite "spellcaster" as an attunement requirement, specifically "requires attunement by a spellcaster" while others list multiple spellcasting classes (e.g., a Staff of Fire "requires attunement by a druid, sorcerer, warlock, or wizard" p. 201). However, all the DMG states in this regard is:

Some magic items require a creature to form a bond with them before their magical properties can be used. This bond is called attunement, and certain items have a prerequisite for it. If the prerequisite is a class, a creature must be a member of that class to attune to the item.¹ (If the class is a spellcasting class, a monster qualifies if that monster has spell slots and uses that class's spell list. [emphasis and footnote added] (p. 136)

The language of the Mage Slayer feat describes a spellcaster as a creature casting or concentrating on a spell.

You have practiced techniques useful in melee combat against spellcasters […] When a creature within 5 feet of you casts a spell … [and] When you damage a creature that is concentrating on a spell … (PHB, p. 168)

From the above, it does not appear that "spellcaster" would necessarily be considered a class or group of classes, so

Are characters that can cast spells, regardless of their class(es), considered spellcasters?

Or, put another way,

Does "spellcaster" (as an attunement prerequisite) mean any creature who can cast a spell or does it mean all classes with the Spellcasting feature?

I can think of five particular cases—innate abilities, two subclasses and two feats:

High Elves know one wizard cantrip (0-level spell) of their choice. Drow can cast the dancing lights cantrip, 1st-level spell faerie fire and 2nd-level spell darkness. Forest Gnomes know the minor illusion cantrip, and Tieflings can cast the thaumaturgy cantrip and 2nd-level spells hellish rebuke and darkness. Also, many monsters have innate spellcasting abilities, which would trigger the Mage Slayer feat.

Starting at 3rd level, both the Eldritch Knight (Fighter Martial Archetype) and Arcane Trickster (Roguish Archetype) gain the ability to cast wizard spells and gain wizard spell slots (potentially) up to 4th-level spells. PHB, pp. 75 & 98. The latter's 17th-level Spell Thief feature also states "you gain the ability to magically steal the knowledge of how to cast a spell from another spellcaster," (Ibid.) which strongly implies that (at least) the Arcane Trickster is a spellcaster, although neither subclass is specifically listed as a prerequisite for any magic item in the DMG.

Eldritch Knight and Arcane Trickster are also found under Multiclassing, Spell Slots in the PHB:

You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, half your levels (rounded down) in the paladin and ranger classes, and a third of your fighter or rogue levels (rounded down)
if you have the Eldritch Knight or the Arcane Trickster feature. (p. 164)

With the Ritual Caster feat, a character chooses a spellcasting class (except paladin or ranger) and gains the ability to cast spells of that class with the ritual tag as rituals. The character immediately gets two 1st-level ritual spells of that class and has the potential to (eventually) learn all of the ritual spells of the chosen class. PHB, p. 169.

With the Magic Initiate feat, a character chooses a particular spellcasting class (except paladin or ranger) and immediately learns two cantrips and one 1st-level spell from the class with no further advancement. PHB, p. 168.

By definition, it seems all of the above would qualify as spellcasters, but I feel spellcaster typically (or maybe traditionally) refers to bard, cleric, druid, paladin, ranger, sorcerer, warlock and wizard.

1. I believe an exception would be a 13th level or higher Thief (Roguish Archetype) due to the Use Magic Device feature. PHB, p. 97.