"Dhamma" here should be written with a capital "D" and means the core teaching or core truth discovered and taught by the Buddha, namely, what is suffering and what is the cessation of suffering. Therefore, what is meant by statement “the one who sees Dhamma sees dependent origination” is one who sees dependent origination sees the Teaching/Truth of the Buddhas.
Bhikkhus, both formerly and now what I teach is suffering and the cessation of suffering.
MN 22
Now it is for one who feels that I teach: ‘This is suffering,’ and ‘This is the origin of suffering,’ and ‘This is the cessation of suffering,’ and ‘This is the way leading to the cessation of suffering.’
AN 3.61
Now this has been said by the Blessed One: “One who sees dependent origination sees the Dhamma; one who sees the Dhamma sees dependent origination.” And these five aggregates affected by clinging are dependently arisen. The desire, indulgence, inclination and holding based on these five aggregates affected by clinging is the origin of suffering. The removal of desire and lust, the abandonment of desire and lust for these five aggregates affected by clinging is the cessation of suffering.’
MN 28
Enough, Vakkali! Why do you want to see this foul body? One who sees the Dhamma sees me; one who sees me sees the Dhamma. For in seeing the Dhamma, Vakkali, one sees me; and in seeing me, one sees the Dhamma.
What do you think, Vakkali, is form permanent or impermanent... suffering...not-self?”— “Impermanent... suffering... not-self, venerable sir.”…—“Therefore … Seeing thus … He understands: ‘… there is no more for this state of being.’”
SN 22.87
The word 'dhamma' can have various meanings, such as 'phenomena' (nature), law of nature (truth; law), duty according to law of nature (practise) and results of practise. In this case, it means 'Truth' or 'Law' (as 'Core Teaching/Doctrine'), as explained in SN 12.20.
“And what, bhikkhus, is dependent origination? ‘With birth as condition, aging-and-death comes to be’: whether there is an arising of Tathagatas or no arising of Tathagatas, that element still persists, the stableness of the Dhamma (dhammaṭṭhitatā), the fixed course/lawfulness of the Dhamma (dhammaniyāmatā), specific conditionality. A Tathagata awakens to this and breaks through to it. Having done so, he explains it, teaches it, proclaims it, establishes it, discloses it, analyses it, elucidates it. And he says: ‘See! With birth as condition, bhikkhus, aging-and-death.
SN 12.20