When can "very" modify a prepositional phrase?

Solution 1:

“Very out of the way”

It is a bit tough to find cases of very modifying individual prepositions, but it is easy to find cases of very modifying entire prepositional phrases as a unit, just as it does other adjectives and adverbs.

• I think it’s very out of character for him.
• Things can be very out of place.
• Or very out of date.
• And very out of the way.
• They are very on top of music and very adamant about what they carry.
• I just really was very on the edge. Very on the edge, for a while.
• He’s very off his rocker today.
• She is really very behind the times, isn’t she?
• You’re not very with it today, are you?

Sometimes you can use it to mean “very much” or “very far”.

• Are you for Obama? Yes, I’m a little bit for him, just not very for him.
• Are you against the draft? Yes, I’m very against it.
• Is that over your head? Yes, but not very over.
• Walk down the street past my house, but not very past it.
• If you can be more into something, you can be very into it. Is there anything you’re less into?
• No thank you, that whole scene is very beneath me.

At which point, the whilom preposition starts acting more adverbially, and adverbs can be veried without half so much trouble. But some of the directional or adverbial ones really do seem to be modifiable by degree:

• Is the end near? Yes, I believe we’re very near the end now.
• Will the parade pass near here? Yes, it will pass very near us. It shall pass very near, indeed.

Being very like something

However, back to your main point about very like something.

I’m not sure that I’m willing to call like a preposition (the OED calls it an adjective for these sorts of uses), but here are assorted OED citations of “very like”:

• 1719 De Foe Crusoe i. xᴠɪ, ― It was very like the Tree we call Fustic.
• 1710 Swift Jrnl. to Stella 25 Oct., ― Addison’s sister is a sort of a wit, very like him.
• 1727 A. Hamilton New Acc. E. Ind. II. xxxɪx. 81 ― The Durean is another excellent Fruit, but offensive to some Peoples Noses, for it smells very like human Excrements, but when once tasted the Smell vanishes.
• 1868 Yates Rock Ahead II. 245 ― Wooded uplands suggested good cover-shooting; broad expanse of heath looked very like rabbits.
• 1857 R. Tomes Amer. in Japan vi. 135 ― The jamana··is very like the red-wood of Brazil and Mexico.
• 1865 Pall Mall G. 25 Oct. 10 ― Gladiateur’s colours are blue and red, and Nu’s are cerise (which is very like red) and blue.
• 1836 Sir G. Head Home Tour 144 ― The town of Dewsbury··celebrated for··grinding old garments into new; literally tearing in pieces fusty old rags··by a machine called a ‘devil’, till a substance very like the original is reproduced.
• 1926 F. Z. Snoop Reproduction & Sexual Evolution 83 ― Havelock Ellis quotes other cases, even butterflies (if insects may be here included) who possess excrescences on their penes, which of necessity must cause pain, or something very like pain, during coition.

And regarding your original Shakespearian citation, the OED notes that this has become (or was at one time) a set phrase, to be very like a whale:

Allusive, proverbial, transf., and fig. uses of sense 1. a. Prov. phr. (to throw out) a tub to the whale: see tub sb.^{9 b.}very like a whale (after Shaks. Ham. ɪɪɪ. ii. 398): see quot. 1859.

• 1591 1st Pt. Troub. Raigne K. John (1611) C 3 b, ― The mariner, Spying the hugie Whale, whose monstrous bulke Doth beare the waues like mountaines fore the wind, That throwes out emptie vessels, so to stay His fury.

• 1859 Slang Dict. 115 ― Very like a whale, said of anything that is very improbable.

So in summary, I think that whatever you call like, there is no problem with things being very like other things.

Solution 2:

M-W.com says of this usage of like, which is an adjective and not a preposition:

3 like [adjective]

1a : the same or nearly the same (as in appearance, character, or quantity) [suits of like design] —formerly used with as, unto, of [it behoved him to be made like unto his brethren — Heb 2:17 (Authorized Version)]

1b chiefly British : closely resembling the subject or original [the portrait is very like]

Very like a whale is perfectly grammatical in contemporary English. There's nothing wrong with it at all.

Similar to is not a phrasal preposition; it's an adjective phrase because, as stated above, like is an adjective and not a preposition. Grammatical terminology is not easy; it's often misunderstood and misused; and it's often disputed by different schools of linguistics. The names of parts of speech, however, are rarely disputed -- except, perhaps, for a few like particle and adverb. What is more important than the part-of-speech label, however, is the function of the word in the sentence.