Yes, melee spell attacks are melee attacks

The rules define melee attacks to include both melee weapon attacks and melee spell attacks (PHB 195):

Used in hand-to-hand combat, a melee attack allows you to attack a foe within your reach. A melee attack typically uses a handheld weapon such as a sword, a warhammer, or an axe. A typical monster makes a melee attack when it strikes with its claws, horns, teeth, tentacles, or other body part. A few spells also involve making a melee attack.

Also, I found a relevant Q&A Twitter post from Jeremy Crawford (the Lead Designer and Managing Editor for D&D 5e) that sheds some light on this. Crawford's tweets are no longer official rulings as of the 2019 Sage Advice Compendium, but his response quoted below is still useful guidance that matches what is clearly stated in the official rules.

Question

Does a "melee spell attack" count as a "melee attack" for Touch of Death?

Answer

A melee spell attack is, indeed, a melee attack and can qualify for the Death cleric's Touch of Death feature.

This would naturally extend to other effects that alter or activate based on melee attacks.

No, the extra damage is only added to one attack on your next turn.

Read the Absorb Elements text carefully:

Also, the first time you hit with a melee attack on your next turn, the target takes an extra 1d6 damage of the triggering type, and the spell ends.

(emphasis added by me)

The spell only provides a benefit during the turn after you cast Absorb Elements. You cannot "store" the extra damage across multiple turns.

If the Sorcerer did not make a melee attack roll on their next turn, then the benefit of Absorb Elements is lost at the end of that turn.

Note: It is possible to cast Absorb Elements twice such that the durations overlap, but the damage won't stack. This is because a character's Reaction resets at the start of their turn, not at the end:

When you take a Reaction, you can’t take another one until the start of your next turn.

For example, suppose the Sorcerer takes fire damage before their turn, using their reaction to cast Absorb Elements. Then on the Sorcerer's turn, once their reaction resets, they somehow take fire damage again, and spend their reaction (again) to cast Absorb Elements (again). This would mean having multiple castings of the same spell, but their benefits do not stack:

The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect--such as the highest bonus--from those castings applies while their durations overlap, or the most recent effect applies if the castings are equally potent and their durations overlap.

If the Sorcerer proceeds to make an attack on this same turn, then only the "most recent" or "most potent" casting of Absorb Elements would apply to that attack.