Assalam Alaykum dear Brother,
Yes, its a form of Dhikr as sister stated above the commandment is from Almighty Himself, Also if we analyze that when we say ALLAHU AKBAR, its for our own benefit and rewards and seeking nearness to ALLAH, where as we all know that ALLAH is the GREATEST already isnt it?
Likewise when we recite Salawat and send Peace and Blessings to our beloved Prophet [saw], it is as commanded by ALLAH [swt] to do it, so that we make seek nearness and be obedient. Where as every Muslim knows that our beloved Prophet [saw] is in the best state with our Lord, and we have to an Emaan on it

Alhamdulillah
Qur’an 33:56;
‘Verily, Allah and His Angels send blessings on the Prophet: O you who believe! Send blessings on him, and salute him with a worthy salutation.’
Allah Almighty says,
"Say [O' Muhammad(saw)]: 'If you love Allah, then follow me and Allah will love you and forgive you for your wrong actions, Allah is Ever-Forgiving, Most Merciful." Qur'an 3:31Muslim narrated in his Sahih from Abu Hurairah (may Allah be pleased with him) that the Messenger of Allah (may Allah bless him and give him peace) said ‘
he who invokes blessings on me once, Allah sends ten blessings upon him.’
Tirmidhi narrated that ‘Umar ibn al-Khattab (may Allah be pleased with him) said
‘Verily, supplications are stalled between the heavens and the earth, and are not lifted up until the supplicant invokes blessings upon the Prophet (may Allah bless him and give him peace).’
Tirmidhi narrated from ‘Amir bin Rani’ah (may Allah be pleased with him) that the Prophet (may Allah bless him and give him peace) said
‘he who invokes blessings upon me, the Angels send blessings upon him equal to that which he invoked, so let the worshipper invoke some, or increase upon that.’
And ALLAH knows the best