No its not, but...
The example you give has a force that comes from it's "wrongness". It leads the listener or reader to consider the meaning behind what are relatively stale metaphors, of the sort that are normally best avoided anyway, and adds the sort of humour that suggests wisdom (rightly or not, people are more prepared to believe something when it's funny).
The advice that one shouldn't mix metaphors isn't a grammar or syntax rule; it's a stylistic guideline.
Generally, if you mix metaphors you run the risk of weakening the metaphors partly because you've done damage to the comparison you were making, and partly because you focus the listener or readers attention on your technique rather than your message.
And since normally you want clear images in your audience's mind, and for them to focus on your message rather than your delivery, mixing metaphors is therefore generally bad.
But, if you make that very reaction part of your rhetorical approach, then it can either drive a message home, make them laugh (a great many humourists are fond of deliberately mixed metaphors for precisely this reason) or both.
So too with anything that brings the mechanics of the metaphor to our attention, rather than let it subtly fo its work, as is normally best. E.g.:
Using a metaphor in front of a man as unimaginative as Ridcully was like a red flag to a bu... was like putting something very annoying in front of someone who was annoyed by it.
That completely destroys the metaphor as we'd normally use it, but it's value is precisely that it did so. With a fresh lively and apposite metaphor this would be a mistake, but here it's the whole point.
In all, there's a good reason not to damage the delivery of a metaphor, and mixing it is one of the ways you can do so, but once you know why you normally shouldn't, you can choose to do precisely that.
Of course with a bad mixed metaphor you can end up doing both; your piece utterly fails as intended, but continues to be repeated as (unintended) comedy. My favourite is one from the House of Commons:
I smell a rat. I see it floating in the air. But I shall nip it in the bud!
Conversely, a deliberate case from the same House was:
The Honorable and Gallant Member is a cowardly liar.
It's not quite a metaphor, since one is meant to believe that Members really are honourable and those in the armed services really are Gallant. But since Parliamentary language requires one to say so whether one believes it or not, that creates a new sense of each that the above deliberately mixes in a way that is literally "bad", as it results in something that taken one way is contradictory and in another a pairing of mismatched senses of which a mixed metaphor is another example (the guideline that one shouldn't mix metaphors a specific case of that) but in effect very good.
(It's still unparliamentary language and would bring censure from the Speaker, but that's another matter).