The Installing of ‘the Priest' and His Assistants (Exodus 40:12-15).
‘And you shall bring Aaron and his sons to the door of the Tent of Meeting, and shall wash them with water. And you shall put on Aaron the holy garments; and you shall anoint him, and sanctify him, that he may minister to me in the priest's office. And you shall bring his sons, and put coats upon them; and you shall anoint them, as you anointed their father, that they may minister to me in the priest's office: and their anointing shall be to them for an everlasting priesthood throughout their generations.'
The installing of the priests follows. Firstly ‘The Priest', and then his assistants in the priestly office. It commences with the washing of water to remove earthiness and physical defilement. Then Aaron is clothed by Moses in the holy garments set apart for ‘The Priest', and Moses anoints him with the holy anointing oil and ‘sanctifies' him, that is, sets him apart to a holy purpose and as belonging to Yahweh.
After this the sons are brought forward and Moses is to dress them in their priestly robes, and anoint them so that they too may minister in the priestly office as assistants to ‘the Priest'. And this anointing is to be seen as setting them and their descendants apart for ever in an everlasting priesthood, as long as the anointing continues. The provision is always that if they prove unworthy the anointing will cease. There are no genuine anointed Aaronic priests today.
In Leviticus 8 only Aaron is specifically anointed on the head (Exodus 40:12), although his sons share in that anointing by the sprinkling on them of the oil (Exodus 40:30). This may be what is in mind here.
The point is not that their anointing will follow the same pattern as their father's, but that they too will be anointed with the holy anointing oil. He was exclusively ‘the anointed Priest' (Leviticus 4:5; Leviticus 4:16; Leviticus 6:22; Leviticus 16:32 etc.). The oil was poured on him (Exodus 29:7; Leviticus 8:12) but only sprinkled on his sons (Exodus 29:21; Leviticus 8:30). Eleazar would have oil poured on him when he later became ‘The Priest'.
However it may be that ‘and you shall anoint them (when you anoint them as Priest), as you anointed their father, that they may minister to me in the priest's office: and their anointing shall be to them for an everlasting priesthood throughout their generations', refers to the future when each Priest will be anointed from among the sons of Aaron on his inauguration as High Priest.
Today it is all who are truly His who are set apart as priests in accordance with His promise to make His people a kingdom of priests (Exodus 19:5-6; 1 Peter 2:5; 1 Peter 2:9), and all, both male and female, are anointed with the Holy Spirit (1 John 2:20; 1 John 2:27; 2 Corinthians 1:21), and sanctified as set apart to a holy purpose and as belonging to Him (1Co 1:2; 1 Corinthians 6:11; Hebrews 10:14). And all are assistants to the One Who uniquely and for ever is ‘The Priest', our great High Priest (Hebrews 9:11; Hebrews 10:21). And that priesthood does not involve a continuation of physical sacrifices, for the final sacrifice was offered once for all in Christ (Hebrews 10:12), but a sacrifice of praise and thanksgiving and dutiful service (Romans 12:1-2; Hebrews 13:15; 1 Peter 2:5), and involves showing forth the excellencies of Him Who has called us out of darkness into His marvellous light (1 Peter 2:9).