Background (3E)
Although the question is tagged as [dnd-5e], Sorcerers have a lower number of spells known since their conception as a Spontaneous Casters in 3E. This was actually already questioned and answered here. The answer links to this amazing interview.
Shane_M: Why do sorcerers have so few spells that they can “know” and does that number increase with Charisma like spells cast does?
TheSage: Three reasons. First, there has to be a hard choice between choosing a wizard and a sorcerer. Second, the sorcerer class, as I mentioned earlier, is for players who don't want to mess around with piles of spells. Finally, the class concept, a “natural” spellcaster, lends itself to a small, personal bag of tricks. Sorcerers do get bonus spells for high Charisma, but that's only for the number of spells they can use each day. Their selection of spells is fixed (though it expands with increasing level).
Also
TheSage: Well, Durst, the fact is that a wizard who finds a scroll can copy that sucker right into her spellbook and the moment she finishes that spell is part or her personal magical arsenal. Sorcerers acquire spells much more slowly.
but this part is more about "that's how it works" than "that's WHY it works like this".
Changes (and some speculation)
However, if you've played earlier editions, you noted that in 5E we got a much easier mechanic of preparing spells.
The next statement is completely speculative: They probably tried to keep the balance between (easier) Prepared and Spontaneous through class features such as metamagic, natural draconic armor and extra spell slots through Flexible Casting than by just giving them a bunch of more known spells. End of completely specutalive I try to back up this speculation with official quotes and my interpretation on them now.
Quick note: When I talk about "balancing", I refer to this sentence:
there has to be a hard choice between choosing a wizard and a sorcerer.
Explaining the Changes
As OP mentioned he started playing at 5E, and other people probably did as well, it is worth explaining the difference between preparing spells in earlier editions (at least, in third edition - I did not play 4E) and now.
Earlier, you had to prepare the spell the exact number of times you would like to use it. If you wanted to cast a fireball twice in a day, you would have to prepare the fireball twice. This way, your actual versality in any given day could be lower than of a Sorcerer's, depending on how you prepared your spells, even if you knew more spells, you might prepare less. If you prepared all different spells, you could use any of them only once. Not a problem for Mage Armor, but a major problem for Fireball if your day was a combat day.
One more additional information that might be relevant is that back in the day, as mentioned in the interview, sorcerers being Charisma-based was actually a downside, since Intelligence was an important stat that defined how many skills you would get. Nowadays Intelligence is arguably the worst stat ever in a non-strongly-role-play setup, while Charisma influences slightly more relevant social skills.
Backing the speculation up
Now, analysing the link to "Modifying Classes" provided by Icyfire:
Sorcerers have the full spellcasting progression; any changes to the Spellcasting feature will have a big impact on the class.
This includes, but is not limited to, the number of spells known, since it is part of the Spellcasting feature. So it should be safe to say that yes, the Sorcerer class was designed to be balanced with that number of spells.
Also, another important part that Icyfire already quoted is
Like bards, sorcerers are have (sic) a limitation on the number of spells they can choose from, which is a major restriction on the class.
Bards know a limited number of spells, which is a restriction on their versatility that should be modified with care when making changes to the class.
Note that Bards still know significantly more spells than Sorcerers, so it's strongly implied that the low versality is intended for balancing, adding to the historical reason.
Actually, I can't say for certain that the historical reason is even relevant. But since historical reason is enough to go against developer's preferences (when added some player pressure), it should have had some impact. (I believe this speculation is based enough)
