Different bells sound different
We know that handbells can sound different than one another. If you were choosing physical handbells for alarms, you could easily choose ones with different pitches or tonal quality.
Since the spell simply says the alarm sounds like “a hand bell,” the caster gets to choose which bell.
Contrariwise, can 2 Alarm spells sound exactly alike?
Let’s turn this question around and ask: Can Alarm create two alarm bells that sound exactly alike? (I’m going to cleverly mislead my enemies with a false alarm.) The spell does not specify that you can do that, either.
If we want to remain consistent, we can’t say, “No, because the spell doesn’t say you can do that,” to both this question and the OP’s. We’ve got to fall back on context, on what actual bells are like.
(Aside: If we would be lenient, we would allow the caster to choose the bell’s ring, be it the same or different than other alarms, because it’s magic. But if we would be strict, we should at least be consistent and reasonable in our strictness.)
Cheap alarm bells naturally sound different
It doesn’t require any special effort to make (mundane) bells that sound different. Moreover, making two bells that sound the same requires tremendous craftmanship. Noted bell manufacturer Jan Felczyński writes:
Bell's acoustic is very complicated matter. The art of creating well-tuned bells is the secret of every bell founder. Other than the pure shape of the bell factors which determine the quality of bell sound are [material, casting, bell's artwork].
If you’ve watched Downton Abbey maybe you’ve noticed the alarm bells that ring for the servants to attend to the gentry all sound different. That’s not hard to accomplish; it is the default case.
If two regular hand bells sound different, magical bells should be able to have that property too. Magical effects should not be assumed to “work like mundane equivalents, only worse.”
There’s no particular reason to make pings different than bells
I’d go a step farther and suggest the “mental ping” would work the same way. The differences between what the mental ping does and the bell does are spelled out, but it doesn't include any differences on whether different alarms can be distinguished.
Thus, if we accept that bells can sound different, we probably ought to accept that "mental pings" (whatever they are) can also be distinguished from one another.
There are limits to this, though. Two alarms could be distinct as two very different bells, but ten or fifty would be harder to keep track of.
Many caster choices are presumed
But why does the caster get to choose the bell sound? Because other spells that allow effects within a category work that way, by default.
Silent Image (PH 296) can create the image of “a creature” (among other things). We assume which creature appears is the caster’s choice, and not some “generic” creature, even though the spell never says the caster can choose which creature.
More examples of spell effects which allow the caster to choose within set parameters include:
- Teleporting to “an unoccupied space” such as in Misty Step (PH 260)
- Choosing “a new form” in Polymorph (PH 266)
- “A target within range”
We understand the caster chooses which form, beast, image, sound, object, space, etc. — unless the spell description says otherwise. The same goes for which bell.
The restriction in the spell is that an audible alarm must sound like a bell, not a roaring tiger, etc. But within that restriction, the caster is free to choose the details.