Verbal components are audible, but the message is not.
The message itself is separate from the verbal component
The message spell has a verbal component. However, the verbal component of the spell is not the same as the message itself, merely a necessary part of casting the the spell in the first place; the actual message is part of the spell effect. (Likewise, the somatic component of message is separate from the caster pointing at the targeted creature as part of the spell's effect.)
This is supported by a related official ruling in the Sage Advice Compendium regarding the spells suggestion and command, both of which also have verbal components:
Is the sentence of suggestion in the suggestion spell the verbal
component, or is the verbal component separate?
Verbal components are mystic words (PH, 203), not normal speech. The
spell’s suggestion is an intelligible utterance that is separate from
the verbal component. The command spell is the simplest example of
this principle. The utterance of the verbal component is separate
from, and precedes, any verbal utterance that would bring about the
spell’s effect.
All three of these spells have a verbal component, and separately involve speaking a word or sentence as part of the spell effect. They all work the same way in this regard; the verbal component that's part of the casting of the spell is distinct from the utterance spoken as part of the spell's effect.
Verbal components are audible, but it's up to the DM how audible
The description of verbal components in the rules states, in part (PHB, p. 203):
Most spells require the chanting of mystic words. The words themselves
aren't the source of the spell's power; rather, the particular
combination of sounds, with specific pitch and resonance, sets the
threads of magic in motion.
The rules don't specify how loud a verbal component has to be, so it is left to DM discretion. This is also reiterated by rules designer Jeremy Crawford in an unofficial tweet from October 2015:
when I cast a spell does the verbal component has to be clearly
audible or can I "whisper" the spell?
The verbal component of a spell must be audible to work. How loud
is audible? That's up to the DM.
The caster ostensibly doesn't have to scream it at the top of their lungs, but in general I think verbal components of spells would need to be spoken in (approximately) a regular speaking voice, or at a comparable volume.
DMs may choose to allow spellcasters to attempt to hide spell components (e.g. by whispering/muttering the verbal component) - potentially requiring an ability check to do so - but this runs the risk of devaluing the sorcerer's Subtle Spell Metamagic option (which removes verbal and somatic components from the spell entirely). Even beyond that, granting all casters a way to hide spell components for free generally makes all spellcasters more powerful regardless.
Only the target hears the message; only the caster hears the reply
As for the volume of the message, the description of the message cantrip states (emphasis mine):
You point your finger toward a creature within range and whisper a
message. The target (and only the target) hears the message and
can reply in a whisper that only you can hear.
The spell description makes it clear how loud the message itself is, and who can hear it. The caster whispers a message that only the target can hear, and the target can reply in a whisper that only the caster can hear. Thus, even if someone can tell that you're casting a spell, only the creature you point at can hear the actual message.
Yes, the target hears the message, even if there are other ambient noises.
The message spell is a magical effect, and doesn't actually create a sound as per normal speech. Instead, only the target hears it.
You point your finger toward a creature within range and whisper a message. The target (and only the target) hears the message and can reply in a whisper that only you can hear.
As you can imagine, this isn't how sound works according to conventional physics. But it's magic, and magic generally doesn't care about the laws of nature.
However, the message spell requires a verbal component to cast. If something prevents the spellcaster from performing the verbal component, then the spell would fail.
If you can’t provide one or more of a spell’s components, you are unable to cast the spell. [...] Thus, a character who is gagged or in an area of silence, such as one created by the silence spell, can’t cast a spell with a verbal component.
If something prevents the caster from whispering the message, then it's likely that they can't perform the verbal components for the message cantrip either.
Best Answer
As you reported, Wild Shape says
Whisper means
A bear can neither speak nor whisper, hence your DM was right in this ruling. Moreover, the Sage Advice Compendium supports your DM's decision:
You can come up with a strategy to overcome this problem: you may decide with the other members of your group a sort of simple code for communicating via Message while you are wildshaped. For example, in the case of your being a bear, you can decide that a low growl means "yes", two quick low growls mean "no" and so on. This is up to you and mainly to your DM.
Another way is to wildshape into a raven (available at 8th level):
Moreover, there are several animals\$^1\$ that speaks their very unique language: Giant Owl, Gian Elk and Giant Eagle. Problem is that the caster of Message should be able to understand those languages.
\$^1\$ Thanks to @Kogarashi Kaito for these further examples.