It is customary for Muslims to congratulate each other on the day of Eid after the prayer has been performed. This was a practice of some of the companions and righteous predecessors after them, and it is very common today.
Muhammad ibn Ziyad reported: I was with Abu Umamah al-Bahili and others among the companions of the Prophet, peace and blessings be upon him. When they would return from Eid prayer, they would say to each other:
تَقَبَّلَ اللَّهُ مِنَّا وَمِنْك
May Allah accept good deeds from us and from you.
Source: Tārīkh al-Kabīr lil-Bukhārī 1/116, Grade: Jayyid
Imam Ahmad commented on this tradition, saying:
وَلَا بَأْسَ أَنْ يَقُولَ الرَّجُل لِلرَّجُلِ يَوْمَ الْعِيدِ تَقَبَّلَ اللَّهُ مِنَّا وَمِنْك
There is nothing wrong if a man says to another man on the day of Eid: May Allah accept good deeds from us and from you.
Source: al-Mughnī 2/295
Jubayr ibn Nufayr reported: The companions of the Messenger of Allah, peace and blessings be upon him, when they met each other on the day of Eid, they would say to each other:
تَقَبَّلَ اللَّهُ مِنَّا وَمِنْكَ
May Allah accept good deeds from us and from you.
Source: Ṣalāt al-‘Īdayn lil-Maḥāmilī 2/218, Grade: Hasan
Adham reported: We would say to Umar ibn Abdul Aziz, may Allah have mercy on him, on the two Eids:
تَقَبَّلَ اللهُ مِنَّا وَمِنْكَ يَا أَمِيرَ الْمُؤْمِنِينَ
May Allah accept good deeds from us and from you, O commander of the faithful.
Hawshab ibn ‘Aqil reported: I met Al-Hasan al-Basri, may Allah have mercy on him, on the day of Eid and I said:
تَقَبَّلَ اللَّهُ مِنَّا وَمِنْكَ
May Allah accept good deeds from us and from you.
Al-Hasan said:
نَعَمْ تَقَبَّلَ اللَّهُ مِنَّا وَمِنْكَ
Yes, may Allah accept good deeds from us and from you.
Source: al-Du’ā’ li-Ṭabarāni 930
Nonetheless, it is important to acknowledge the status of such phrases in Islam. It is certainly praiseworthy to supplicate to Allah on behalf of yourself and others. This particular formula and others like it, however, is not directly from the Sunnah. It is merely permissible to say and good in so far as it is a supplication to Allah.
Ibn Taymiyyah was asked specifically about this phrase, as well as other customary supplications, and he responded, writing:
أَمَّا التَّهْنِئَةُ يَوْمَ الْعِيدِ يَقُولُ بَعْضُهُمْ لِبَعْضٍ إذَا لَقِيَهُ بَعْدَ صَلَاةِ الْعِيدِ تَقَبَّلَ اللَّهُ مِنَّا وَمِنْكُمْ وَأَحَالَهُ اللَّهُ عَلَيْك وَنَحْوُ ذَلِكَ فَهَذَا قَدْ رُوِيَ عَنْ طَائِفَةٍ مِنْ الصَّحَابَةِ أَنَّهُمْ كَانُوا يَفْعَلُونَهُ وَرَخَّصَ فِيهِ الْأَئِمَّةُ كَأَحْمَدَ وَغَيْرِهِ لَكِنْ قَالَ أَحْمَد أَنَا لَا أَبْتَدِئُ أَحَدًا فَإِنْ ابْتَدَأَنِي أَحَدٌ أَجَبْته وَذَلِكَ لِأَنَّ جَوَابَ التَّحِيَّةِ وَاجِبٌ وَأَمَّا الِابْتِدَاءُ بِالتَّهْنِئَةِ فَلَيْسَ سُنَّةً مَأْمُورًا بِهَا وَلَا هُوَ أَيْضًا مِمَّا نُهِيَ عَنْهُ فَمَنْ فَعَلَهُ فَلَهُ قُدْوَةٌ وَمَنْ تَرَكَهُ فَلَهُ قُدْوَةٌ وَاَللَّهُ أَعْلَمُ
As for congratulations on the day of Eid that people say when they meet each other after the Eid prayer, ‘May Allah accept from us and from you,’ and, ‘May Allah continue to bless you,’ and so on, these have been narrated from a group of the companions that they would do so, and it is a concession granted by the Imams such as Ahmad and others. Rather, Ahmad said, ‘I do not initiate it with anyone, but if anyone initiates it with me, I will answer him.’ That is because an answer to a greeting is obligatory. As for initiating the congratulations, it is not a sunnah that has been commanded, nor has it been prohibited also. Whoever does so has a precedent, and whoever leaves it has a precedent. Allah knows best.
Source: Majmū’ al-Fatāwá 24/253
Thus, it is commendable for Muslims to congratulate each other on the day of Eid by asking Allah to accept all of our good deeds. It is virtuous because it is a supplication to Allah, and supplication is the essence of worship. On the other hand, this specific phrasing has been narrated from some of the companions and not the Prophet (ṣ) himself, which means it is good and permissible, but not strictly part of the Sunnah or an obligation to say it.
Success comes from Allah, and Allah knows best.