Maa Sita and Maa Rukmini are fundamentally the same soul—Mahalakshmi. The way Lord Vishnu does not change His core divine essence whether He walks the forests as Rama or plays the flute in Mathura as Krishna, His eternal consort—Mahalakshmi—descends as Sita in the Treta Yuga and as Rukmini in the Dvapara Yuga.
Similarities:
1. Both are Para Lakshmi(Shri)/Para Shakti
एक एव परमा लक्ष्मी प्रमोदवनचन्द्रिका।
सहजानन्दानी साक्षाद् रमते रामसंगिनी।।
She(Sita) alone is the Supreme lakshmi—the cool, soothing moonlight illumination of the garden of supreme bliss(pramodvan) . Bestowing inherent divine joy, She directly revels as the eternal, inseparable companion of Lord Rama.
—Adi Ramayana, chapter 3
ङ्य लक्ष्मीः पराशक्तिर्विश्वानुग्रहरूपिणी ।
That Rukmini is Lakshmi, the supreme transcendental power (Para-shakti), whose very form is the embodiment of grace and compassion showered upon the entire universe.
—Gautamiya mahatantra
2. Identify as the Primordial Nature (Mulaprakriti)
मूलप्रकृतिरूपत्वात्सा सीता प्रकृतिः स्मृता ।
Being the first cause(Mulprakriti), Sita is known as Prakriti.
—Sita Upanishad, Atharva Veda
कृष्णात्मिका जगत्कर्त्री मूलप्रकृती रुक्मिणी ॥
"Rukmini, the Queen of Sri Krishna is Krishnatmika (soul of lord Krishna), having Krishna as her essence, Rukmini is the basis of the material nature (Mulaprakriti) and the creator of the whole universe.
—Gopal Uttar-tapani upanishad, Atharva Veda
3.Personification of the Sacred Syllable Om (Pranava)
प्रणवप्रकृतिरूपत्वात्सा सीता प्रकृतिरुच्यते ।
Sita is known as Prakriti of Pranava, She is cause and so is named Prakriti.
—Sita Upanishad, Atharva Veda
प्रणवत्वेन प्रकृतित्वं वदन्ति ब्रह्मवादिनः ॥ १३॥
As a result of her association with Brahman, the Brahmavadins identify Krsna's Prakrti (Rukmini) with Pranava.
—Gopal Uttar-tapani upanishad, Atharva Veda
4. Both are considered as the Mul-Avtari Mahalakshmi herself who is the source of any other incarnations and accompanied lord Viṣṇu in every incarnations.
यदायदा जगत्स्वामिन्देवदेव जनार्द्दन ॥ १७ ॥
अवतारान्करोषि त्वं तदेयं त्वत्सहायिनी ।
यदा त्वं भार्गवो रामस्तदाभूद्धरणी त्वियम् ॥ १८ ॥
अधुना जानकी जाता भवित्री रुक्मिणी ततः ।
अन्येषु चावतारेषु विष्णोरेषा सहायिनी ॥ १९ ॥O Janārdana, O lord of Devas, O lord of the universe, whenever you take up incarnations, she too assists you. When you were Bhārgava Rāma, she had been Dharaṇī (Earth). Now she has become Jānakī and she will be born as Rukmiṇī. In other incarnations too she assists Viṣṇu.
—Skanda Mahapuran, Brahma Khanda, Setu-Mahatmya, Chapter 22
राघवत्वेभवत्सीता रुक्मिणी कृष्णजन्मनि ।
अन्येष्वेवावतारेषु विष्णोरेषा सहायिनी ॥ १५ ॥She was Sītā in (the lord’s) existence as Rāma, and now Rukmiṇī in his birth as Kṛṣṇa. She accompanied Viṣṇu in his other incarnations also.
—Padma Mahapuran 6.247
5. Tri-guna-atmika, the Embodiment of the sattva, rajas and tama guna and The threefold Creative Powers like creation sustenance and dissolution.
उद्भवस्थितिसंहारकारिणीं क्लेशहारिणीम् ।
सर्वश्रेयस्करीं सीतां नतोऽहं रामवल्लभाम् ॥ ५ ॥I bow to Sita the beloved consort of Sri Rama, who is responsible for the creation, sustenance and dissolution (of the universe), removes afflictions and begets all bleneshess.
—Sri Ramacaritamanasa, Bala Kanda, Chapter1
देवो ह्यसौ परं ब्रह्म सदाऽनिर्विण्णमानसः ।
मायाशक्तिस्त्वमेतस्य सर्गस्थित्यन्तकारिणः ॥ २२ ॥He (Lord Kṛṣṇa) is indeed the Supreme Brahman, whose mind is eternally blissful and free from sorrow. And You (Devī Rukmiṇī) are His Māyā-Śakti (divine cosmic creative power), through which the creation, preservation, and dissolution of the universe are brought about.
—Skanda Mahapuran 7.4.3
6. Both are considered as Bhagvati(the Saguna Brahman) who possesses all six attributes completely
सीता भगवती ज्ञेया मूलप्रकृतिसंज्ञिता ॥
Sita should be known as Bhagavati, who is designated as Mulaprakriti (the Primordial Cosmic Nature).
—Sita Upanishad, Atharva Veda
त्वं हि साक्षाद्भगवती कृष्णश्च पुरुषोत्तमः ।
You, Rukmini indeed the Supreme goddess Bhagavati herself and Lord Krishna the Supreme Puruṣottama.
—Skanda Mahapuran 7.4.3
7. Both are the Soul of Narayana
सीतारामत्मकं सर्व सर्वकारण कारणम् ।
All this is of the nature of Sita-Rama, the supreme cause of all causes.
—Brihad Vishnu Puran, Mithila Mahatmya, chapter 13
अद्याहमवगच्छामि सर्वथा सर्वशोभने।
आत्माऽद्वितीयः कृष्णस्य भोजे त्वमिति भामिनि॥ ४२O beautiful and glorious daughter of Bhoja (Rukmini), I know you to-day as the soul of the peerless Krishna.
—Harivamsa, Vishnu parva, chapter 66
8. Being the Anapayini Shakti of lord both has Non-Dual Inseparability from the Supreme
अनन्या हि मया सीता भास्करेण प्रभा यथा॥
Indeed, Sita is inseparable from me, just as light is inseparable from the Sun.
—Valmiki Ramayana,6.118.19
त्वंविद्याऽहंपरब्रह्म त्वंमायाचेश्वरस्त्वहम् ।
त्वंबुद्धिरहंजीवो वियोगः कथमावयोः ॥ ६५ ॥Lord Krishna said : O Rukmini, If you are wisdom, I am the supreme consciousness. I am the God, if you are illusion. As you are the wisdom, I am the living being. So how can there be separation between both of us.
—Skanda Mahapuran,Prabhas khanda, Dwaraka mahatmya, chapter 3
9. Both are the ultimate refuge of this entire universe
सद्यस्ते सिद्धिमायांति ये सीतापदचिंतकाः ॥ ३२ ॥
यस्याः संकल्पमात्रेण जन्मस्थितिलयादिकाः ।Those who reflect upon Sītā’s feet, instantly attain perfection; by the mere thought about her, these divine functions of the worlds like origin, maintenance and destruction, take place.
—Padma Mahapuran 5.66
तां रूपिणीं श्रीयमनन्यगतिं निरीक्ष्य ।
या लीलया धृततनोरनुरूपरूपा ।Looking upon Rukmini —the personification of Goddess Śrī, who is the refuge of all and whose form playfully matched His own—Lord Hari, feeling pleased and smiling.
—Srimad Bhagavatam 10.60