There is no exegetical warrant necessary here: the meaning "having the same nature" is perfectly in line with ordinary Ancient Greek usage.
One of the principal meanings of the noun πάθος is "state" or "condition," and we already find Plato and Aristotle using the word in the slightly more general sense denoting a "property" or "quality" of something. Compare Aristotle's Metaphysics (1022b15-20, transl. by Hugh Tredennick):
πάθος λέγεται ἕνα μὲν τρόπον ποιότης καθ᾽ ἣν ἀλλοιοῦσθαι ἐνδέχεται, οἷον τὸ λευκὸν καὶ τὸ μέλαν, καὶ γλυκὺ καὶ πικρόν, καὶ βαρύτης καὶ κουφότης, καὶ ὅσα ἄλλα τοιαῦτα: ἕνα δὲ αἱ τούτων ἐνέργειαι καὶ ἀλλοιώσεις ἤδη. ἔτι τούτων μᾶλλον αἱ βλαβεραὶ ἀλλοιώσεις καὶ κινήσεις, [20] καὶ μάλιστα αἱ λυπηραὶ βλάβαι. ἔτι τὰ μεγέθη τῶν συμφορῶν καὶ λυπηρῶν πάθη λέγεται.
"Affection" [πάθος] means: (a) in one sense, a quality in virtue of which alteration is possible; e.g., whiteness and blackness, sweetness and bitterness, heaviness and lightness, etc., (b) the actualizations of these qualities; i.e. the alterations already realized, (c) more particularly, hurtful alterations and motions, [20] and especially hurts which cause suffering, (d) extreme cases of misfortune and suffering are called "affections."
This quote is particularly relevant because it demonstrates how in the Greek language πάθος could denote both the "subjective" suffering that we mostly associate with the words "pathos" and "pathetic," but also the more "objective" notion of "quality" or "property." Note how Aristotle actually even puts the objective meanings before the subjective ones.
If we start from the Aristotelian definition of πάθος, then, the word ὁμοιοπαθής could be used to mean either "having the same qualities, properties, nature" or "having the same passions, emotions" (which I think is closer to what you have in mind). And these are indeed the two meanings of ὁμοιοπαθής given by the Liddell-Scott-Jones dictionary.
Edit: As Susan rightly points out, in what I wrote I assumed more or less that the NT author would have analyzed ὁμοιοπαθής as a compound, so that we could reduce the question of its meaning to the question of the meaning of its constituent words, especially that of πάθος. Seeing as how many compounds involving ὁμοιο- there are, and how scanty the attestation of ὁμοιοπαθής seems to be according to the dictionary, I still think this is likely, but let me give two usage examples (from Attic Greek) to support my main point.
From Theophrastus' Enquiry into Plants (5.7.2, transl. by Sir Arthur Hort):
However oak-wood does not join well with glue on to fir or silver-fir; for the one is of close, the other of open grain, the one is uniform, the other not so; whereas things which are to be made into one piece should be of similar character, and not of opposite character, like wood and stone.
The above is clear enough. The following example, from Plato's Timaeus (45c,d) is perhaps less clear, since it is a quite involved explanation of the mechanism behind human vision. However, it still brings home the point that ὁμοιοπαθής can be used to refer to "objective" properties, rather than "subjective" emotions or experiences. Two translations are given to make the passage more understandable (transl. 1: W.R.M. Lamb, transl. 2: Robin Waterfield):
ὅταν οὖν μεθημερινὸν ᾖ φῶς περὶ τὸ τῆς ὄψεως ῥεῦμα, τότε ἐκπῖπτον ὅμοιον πρὸς ὅμοιον, συμπαγὲς γενόμενον, ἓν σῶμα οἰκειωθὲν συνέστη κατὰ τὴν τῶν ὀμμάτων εὐθυωρίαν, ὅπῃπερ ἂν ἀντερείδῃ τὸ προσπῖπτον ἔνδοθεν πρὸς ὃ τῶν ἔξω συνέπεσεν. ὁμοιοπαθὲς δὴ δι᾽ ὁμοιότητα πᾶν γενόμενον, ὅτου τε ἂν αὐτό ποτε ἐφάπτηται καὶ ὃ ἂν ἄλλο ἐκείνου, τούτων τὰς κινήσεις διαδιδὸν εἰς ἅπαν τὸ σῶμα μέχρι τῆς ψυχῆς αἴσθησιν παρέσχετο ταύτην ᾗ δὴ ὁρᾶν φαμεν.
So whenever the stream of vision is surrounded by midday light, it flows out like unto like, and coalescing therewith it forms one kindred substance along the path of the eyes' vision, wheresoever the fire which streams from within collides with an obstructing object without. And this substance, having all become similar in its properties because of its similar nature, distributes the motions of every object it touches, or whereby it is touched, throughout all the body even unto the Soul, and brings about that sensation which we now term “seeing.”
So whenever the ray that flows through the eyes issues forth into surrounding daylight, like meets with like and coalesces with it, until a single, undifferentiated stuff is formed, in alignment with the direction of the eyes, wherever the fire from inside strikes and pushes up against an external object. The similarity between the fire from within and the fire outside means that the stuff is completely homogeneous, and whenever it touches or is touched by anything else, it transmits the object's impulses right through itself and all the way up to the soul, and the result is the perception we call 'seeing'.
Brief conclusion: Theophrastus' passage couldn't be clearer: he uses ὁμοιοπαθής in an "objective" way, to refer to properties of certain types of material (moreover, the properties in question are "static," and not overtly the direct result of change, or of being "affected" in some way -- contra LSJ). On the other hand, Plato's text is more obscure, but it is still sufficiently clear that he is not referring in the least to emotions, but rather to objective properties of substances of some kind.
