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I wonder, can something be both a type and a token (in reference to the type-token distinction in philosophy)? For example, an individual dog is a token of the type of dog, but the type of dog is itself a token of the type of animal species. So, can some entity be both a type and a token? Also, am I misunderstanding the notion of type and token when I mentioned my dog/animal example?

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It seems the answer is depends on whose entries you prefer. According to WP, there are two distinct senses of type-token relationships. But according to SEP, the first sense included in the WP is better called an occurrence rather a token. From Types and Tokens (SEP):

The distinction between a type and its tokens is an ontological one between a general sort of thing and its particular concrete instances (to put it in an intuitive and preliminary way). So for example consider the number of words in the Gertrude Stein line from her poem Sacred Emily on the page in front of the reader's eyes:

   Rose is a rose is a rose is a rose.

In one sense of ‘word’ we may count three different words; in another sense we may count ten different words. C. S. Peirce (1931-58, sec. 4.537) called words in the first sense “types” and words in the second sense “tokens”

The first is separating universals and particulars, and the second is distinguishing repetitious uses of symbols.

Universals and Particulars

According to WP "type-token distinction":

The sentence "they drive the same car" is ambiguous. Do they drive the same type of car (the same model) or the same instance of a car type (a single vehicle)? Clarity requires us to distinguish words that represent abstract types from words that represent objects that embody or exemplify types. The type–token distinction separates types (abstract descriptive concepts) from tokens (objects that instantiate concepts).

Here, we can do the same with 'dog'. They own the same dog. Do they own the same type of dog (the same breed) or the the same instance of a breed (a single physical dog named Fido). What the type-token distinction shows is that there is the abstraction of dogs and breeds and there there are dogs. Again with 'animal' and then we'll see what we see. They own the same animal. Do the own the same type of animal (the same species) or the same instance of an animal (a single physical animal named Fido).

Now what you're asking is, if Fido is both the dog which is the animal, what is the exact relationship. That's a fair question. Under type theory, an instance of a dog is certainly a type of animal, where the set-theoretic relationship is element-subtype-type. But the according to WP, the point of the type-token dichotomy is to draw a distinction between the abstraction and concrete particular. So, if you one accepts that dog is an abstraction of instances of dog, but that dog is also an instance of an abstraction of animals, then yes, a type can be a token given proper context.

Instantiations of Strings and Other Symbols

Later in WP we read:

The word 'letters' was used three times in the above paragraph, each time in a different meaning. The word 'letters' is one of many words having "type–token ambiguity". This section disambiguates 'letters' by separating the three senses using terminology standard in logic today.

This example From C.S. Pierce differs in that it is mapping a string to a meaning. In this case, we are dealing with what Ogden and Richards in their The Meaning of Meaning symbol-referent (where referent is often taken to be an physical object) or what Frege called reference-sense where reference is the syntax that points to the semantics. In this example, "one uses 'letters' to write 'letters' if one is a woman of 'letters'", we have have the same token/symbol/reference/syntax to three different types/referents/senses/meanings.

Now, can we still maintain the token-type dichotomy if the spirit of this example is honored? Well, no, because the type token distinction is about distinguishing occurrences of strings in this sense. For example, we can rewrite the example above as:

One uses letters1 to write letters2 if one is a woman of letters3.

Here, according to Pierce's usage, 'letters' is the type, and 'letters1', 'letters2', and 'letters3' are the tokens.

Which Usage?

I've taken a look at the Encyclopedia of Philosophy, and type-token distinction doesn't have an entry (Volume 10, p.646). It does occur in the index and points to one article which says:

the nominalist must, to be consistent, go further and recognize that what he says of things, if true of them, must be true of words also, which requires him to make what logicians have called the "type-token" distinction. Any occurrence of the word red is individual("red" as a token)k, and two occurrences of what would be called the same word ("Red" as a type) are occurrences of the same word only in that they resemble each other in the relevant ways, Thus the universal word Red becomes the class of the resembling individual words "red," "red," "RED," and so on, and once the universality of the word has been analyzed along these lines, the reason for saying that only words are universal is gone...

This last example of course is referred to as allography in linguistics these days.

So, in the sense of allography, a dog could not be a token of animal, because 'dog1' and 'dog2' are not a concrete instances of 'animal'. It looks like the question of whether a particular philosopher accepts a distinction between tokens and occurrences is a function of one's metaphysical presuppositions in regards to things like realism and nominalism. From WP:

In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels.1 There are at least two main versions of nominalism. One version denies the existence of universals—things that can be instantiated or exemplified by many particular things (e.g., strength, humanity). The other version specifically denies the existence of abstract objects—objects that do not exist in space and time.3

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Consider the following "I came I saw I conquered". How many words are in it? In one sense, there are 4 word types, in another sense there are 6 word tokens.

The type-token distinction is (and originally, dating back to Pierce, if memory serves) best applied to strings/formulas/words. The typical type-token distinction is then as follows: a token is the actual concrete particular composed of light/sound/pixels,etc and the type is the abstract object. Seen thus, it is not possible for something to be both a type and token (since abstract and concrete are generally seen to be mutually exclusive)

Two other notes: (a): tokens are not occurences! The line type contains "instances" of I, but being an abstract object cannot be composed of tokens which are concrete particulars. So we say that the word type "I" occurs in it instead. (see Wetzel for more ) (b): You are likely looking for a distinction somewhere between properties, relations, and/or types. (The types of a type theory, that is).

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You've shifted your own goalposts halfway through. To reframe what you said:

A - Five years ago, Frank's son was born. He named him Thomas. They laughed and played. But two weeks ago, the son died.
B - Oh, how did Thomas die?
A - Thomas didn't die. Frank died.
B - But you said the son died?
A - Yes, because Frank is Albert's son.

Mid explanation, the reference frame of paternity changed. Your understanding of who "the father" and "the son" refers to hinges on this frame. Initially, Frank (father) and Thomas (son) were in focus; but then it shifted to Albert (father) and Frank (son). This renders the explanation hard (if not impossible) to reasonably understand.

Your interpretation of tokens and types is similar. Initially, you're comparing a dog breed (type) and a specific dog of that breed (token), but then you shift into comparing the genus (type) and a breed that falls under that genus (token).

We can keep shifting these goalposts. The genus of dogs (i.e. canis) falls under the family of Canidae, which in turn falls under the order of Carnivora, which in turn falls under the phylum of Chordata, which in turn falls under the class of Mammalia, which in turn falls under the kingdom of Animalia, ...

You need to set a frame of reference in order to make it clear on what level your information is operating, before you can understand what is being referenced.

For example, a children's book may refer to "the dog" as being one of many kinds of animal. Here, "the dog" is a token of the type "animal". But another children's book may be telling a narrative story where all of its charracters are dogs. Here, "the dog" is the overarching type of the individual characters in the story (tokens).

So, can some entity be both a type and a token?

Yes, but not in the same frame of reference. A son can also be a father, but someone cannot be their own father or son.

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