1. The Father and the Son are the same God.
2. For any x and y, and for any kind F, if x and y are the same F, then x is an F, y is an F, and x = y. (x and y are numerically one)
3. The Father = the Son. (1, 2)
(2), however, is false, unless we are making a question-begging assumption, in which case there seems to be an equivocal middle term. To say that something is 'the same F', we usually only require some kind of equivalence relation (a relation that's symmetric, reflexive, and transitive). But identity (represented here as x = y) is only one kind of equivalence relation, namely, the kind with antisymmetry. (Strictly speaking, adding antisymmetry gets you equality -- hence the symbolism. It is in fact not entirely certain that equality and identity are the same relation, since there are accounts you can give of mathematical equality, which is paradigmatic equality, preserving its character as an equivalence relation with antisymmetry, that at least make it seem weaker than what one would want from identity; but this is a contentious issue, and there is no widely accepted view about what you could even conceivably add to equality to make it identity, and people do in general make the assumption that equality is identity, or close enough. It need not make a difference here, since the primary issue turns on identity being an equivalence relation with at least antisymmetry; I mention it only because Tuggy regularly talks as if identity were straightforward rather than something for which there are still many unresolved puzzles. When working with identity, it's wise to go slowly.)
Thus (2) is false in the senses in which we usually talk about things being the same F. If we assume specifically that we are including antisymmetry in 'the same F', then (2) becomes a tautology. But in general the only reason you would ever assume that 'the same F' implies antisymmetry is if you were deliberately doing it in order to get something like (3).
This doesn't even get into the problem of identity across modal domains. Usually when talking about identity we are talking about extensional identity. But we use forms of identity that are not obviously extensional. For instance, if I see someone wearing a hat and then later not wearing a hat:
1. That man with the hat and that man without a hat are the same person.
2. For any x and y, and for any kind F, if x and y are the same F (assuming antisymmetry), then x is an F, y is an F, and x = y.
3. That man with the hat = that man without a hat.
From which you can derive a contradiction (one person being hatted and not hatted), of course, unless one modulates the identity using modal information (that of difference in time). But obviously we do also see immediately that despite being the same person, that man with the hat and that man without the hat differ in properties. Obviously, time is not the only modality that adds this sort of complication. There is no generally accepted account of how to handle identity across modal domains. The three kinds of identity across modal domains most discussed these days are personal identity (i.e., one form of identity through time) and transworld identity (i.e., identity across different possible worlds), and material constitution (if one takes material constitution to be an identity relation; in which case it can involve several different kinds of modal domains). All of them raise remarkably complicated questions, and there is no consensus on the best way of handling any of them.
All three of these, however, require us to recognize that (3) is consistent with x and y also being very different, unless we assume that x and y are not in different modal domains. If they are in different modal domains (different times, different locations, different possibilities, different roles), x can equal y and yet differ from it in quite a few ways (hatted, unhatted; 13,148 days and nine hours old, 13 148 days and ten hours old; etc.). It's pretty clear in context that this is a problem for what Tuggy wants to say, since the point is to press a contradiction on
Bowman rather than just giving a slightly less specific statement of (1), but contradictions can be blocked by difference in modal domain. (This is why the principle of noncontradiction is usually stated as something like 'A cannot be both B and not-B in the same respect', i.e., in the same modal domain.) All of the traditional descriptions of the Trinity, however, and most of the modern 'models', lay out the doctrine in heavily modalized terms, so one would have to rule out, and not merely assume, the possibility that we have different modal domains.
ADDED LATER: James Chastek discusses a counterexample to (2) that is of particular relevance to the question.