Not an answer for USA but for UK.
The National Motor Museum carries this:
What was the first motor car to run on the British Highway?
It is now generally accepted that the Hon. Evelyn Ellis was the first when he imported a Panhard et Levassor into Britain in June 1895 . . .
By the end of 1895, following further importations, it was estimated that there were 14 or 15 cars on Britain’s roads – a figure which had increased dramatically by 1900 to approximately 700 – 800!
Before the automobile, the mechanised road transport of the 19th century was steam powered traction engines. These were legislated by various Locomotive Acts such as the requirement for a man carrying a red flag to walk in front.
The first three, the Locomotives on Highways Act 1861, The Locomotive Act 1865 and Highways and Locomotives (Amendment) Act 1878, contained restrictive measures on the manning and speed of operation of road vehicles; they also formalised many important road concepts such as vehicle registration, registration plates, speed limits, maximum vehicle weight over structures such as bridges, and the organisation of highway authorities.
The Locomotives Act 1898 contains these clauses:
(1) All locomotives not required to be licensed under this Act shall be registered in the county in which they are ordinarily used or to be used in such manner as the county council may direct.
(2) The county council may charge such a fee not exceeding two shillings and sixpence for registration under this section as they think fit, and on registration shall provide the person applying for registration with a plate with the registered number marked upon it.
(3) The plate shall be fixed in a conspicuous position to the locomotive in respect of which it is provided, and shall not be removed without the consent of the council by whom the locomotive is registered.
So, in UK road vehicles in the 19th century carried registration plates before the arrival of the automobile.