English nouns do not have grammatical gender. But in Moby Dick, some nouns do seem to have gender, like "ship" (feminine) and "whale" (masculine). Some passages:
And now the time of tide has come; the ship casts off her cables.
Nor was it his unwonted magnitude, nor his remarkable hue, nor yet his deformed lower jaw, that so much invested the whale with natural terror (...).
Is this some kind of non-standard variety of the language?
Neither of the examples are particularly non-standard.
Like people, sexed animals can also be referred to in either the masculine or feminine as appropriate when the sex is either known or assumed. You wouldn't refer to all whales in the masculine, but in the passage you quoted, the whale itself is a male, and therefore uses masculine pronouns to reference it.
I'll let the question aedia linked handle referring to ships in the feminine more specifically, but basically, it falls into a special case of non-living nouns that have historically been referred to using a feminine form.
It has always been customary to personify certain inanimate objects and attribute to them characteristics peculiar to living creatures. Thus, things without life are often spoken of as having a sex. Some objects are regarded as masculine. The sun, winter, and death are often personified in this way. Others are regarded as feminine, especially those things that are dear to us. The earth as mother Earth is regarded as the common maternal parent of all life. In languages that use gender for common nouns, boats, ships, and other vehicles almost invariably use a feminine form. Likewise, early seafarers spoke of their ships in the feminine gender for the close dependence they had on their ships for life and sustenance.
Male and female animals are referred to by their genders, if known. There is a difference, for example, between a billy goat and a nanny goat. Or a rooster and a hen. It is common to refer to a male animal as "he" or a female as "she" — in fact, where no gender word exists, the 3rd-person singular is prefixed to the word: "she"-ape, etc.
As other answers note, this is not non-standard. The gendering of nouns in English tends to go with the extent to which we personify or humanise them.
Thus, inanimate objects are usually it, as are animals that are not well-known enough to be considered with personalities, even if their sex is known*. Conversely, humans are almost invariably he, she, as are closely-known individual animals (eg the whale in Moby-Dick), and inanimate objects that are highly anthropomorphised — often ships, sometimes countries and planets, occasionally cars, musical instruments, etc.
The OED, for instance, under she, lists the relevant sense as:
2. Used (instead of it) of things to which female sex is conventionally attributed.
2a. Of a ship or boat. Also (now chiefly in colloquial and dialect use), often said of a carriage, a cannon or gun, a tool or utensil of any kind; occas. of other things.
2b. Of abstractions, etc. personified as feminine; also of the soul, a city, the church, a country, †an army, etc.
†2c. rarely Of an immaterial thing without personification.
This corresponds in some ways to the grammatical distinction of animate/inanimate found in many languages (eg Russian); but those terms are a little misleading here, since as mentioned above, animals are very often it in English.
I should add that this is is mostly my own analysis; I don’t know a formal treatment of this in English grammar, and would love to see one. But I’d contend that this — rather than a hangover or influence from more ubiquitous grammatical gender — is the reason why ships and whales are she.
* Compare the Ngrams hits for bull with its vs bull with his, etc; note that most hits with his are references to John Bull, Sitting Bull, or similar.
