On a lot of aircraft, the model number / aircraft name is clearly marked somewhere on the fuselage, e.g. "Airbus A320". Is there any rule or mandate, that all aircraft must bear the model number at some designated place on its body, to help others identify them?
Every aircraft I have flown had a data plate mounted somewhere in the outside, showing the make, model, serial number, and the year of manufacture. You probably can't see it unless you're 3 feet away, but it's there.
The relevant regulation in the US is:
§ 45.11 Marking of products.
(a) Aircraft. A manufacturer of aircraft covered under § 21.182 of this chapter must mark each aircraft by attaching a fireproof identification plate that—
(1) Includes the information specified in § 45.13 using an approved method of fireproof marking;
(2) Must be secured in such a manner that it will not likely be defaced or removed during normal service, or lost or destroyed in an accident; and
(3) Except as provided in paragraphs (d) through (h) of this section, must be secured to the aircraft fuselage exterior so that it is legible to a person on the ground, and must be either adjacent to and aft of the rear-most entrance door or on the fuselage surface near the tail surfaces.
§ 45.13 Identification data.
(a) The identification required by § 45.11 (a) through (c) must include the following information:
(1) Builder's name.
(2) Model designation.
(3) Builder's serial number.
(4) Type certificate number, if any.
(5) Production certificate number, if any.
Here's an example from a Stinson 108:
No. It's easy enough to find photographs online of aircraft that don't have the model number written on them in an obvious way. If there were any regulatory requirement to have the model number painted on the plane for recognition purposes, it would have to be obviously visible or there'd be no point.
FAA AC-45-4 act is regulating that. Each airplane made after that year must have nameplate attach to the fuselage.