The mobile emitter came from a ship that was 6 meters in length and only had a single crewmember. Why would it have a holographic emitter on board?
From the Memory Alpha Article on the mobile emitter:
Owned by Henry Starling in 1996, the emitter was either aboard the appropriated Aeon timeship, or derived from the 29th-century technology therein. Starling used the emitter to grant mobility to his captive, The Doctor, while the latter was displaced in time after he hacked into Voyager's computers and stole the hologram, as equipping The Doctor with the emitter gave Starling the ability to use him as a hostage by threatening to destroy the emitter as it held The Doctor's program. After being returned to the 24th century by Captain Braxton, the USS Voyager was allowed to retain the anachronistic mobile emitter, granting The Doctor a new-found mobility and utility beyond Voyager's sickbay and holodecks, able to slip the emitter on and off relatively easily.
So, it is not clarified where the mobile emitter comes from. I personally prefer the aboard timeship theory. The reason is the following:
From the artilce on Futures End, Part 1
Braxton explains that he was trapped on Earth thirty years ago, in 1967, when his timeship crashed in the mountains. He initiated an emergency beam out but was unable to reach his ship in time and instead it was found by Henry Starling, who began exploiting its 29th century technology, beginning the microcomputer revolution of the 20th century on Earth. Braxton believes it's too late and that "the future's end" is inevitable. He explains the paradox he believes will occur.
So, in the 24th and 29th century, they are still aware of the microcomputer revolution that began in the 20th century with Starling. So, he designed products that he was able to sell. Would the mobile emitter have been something he was able to design, Voyager would've known the design and be able to reproduce it. So, IMHO, Starling has not been able to reproduce the Mobile Emitter. (I am aware that the design could've been forgotten during WW3)
Now, why does the Timeship have a mobile emitter on board. Assume you travel alone around in time. At some point, you might run into a problem and no way to reach anybody from your timeframe. The mobile emitter is an easy an convenient way to have all sorts of specialists (doctor, engineer, hunter (should you crash and need food) and so on) at your disposal at all time. So, it makes perfect sense for a ship to have such a piece of equipment with them.
The mobile emitter may have been intended as a holographic disguise. One aspect of many time-travel missions is the need to blend in when you get there - see for instance "Time's Arrow", "Trials and Tribble-ations", or First Contact. A holoemitter with enough computing power to sift through Federation records and come up with a disguise on its own would save a lone agent time and effort when preparing for a mission to the past, and it would be easier to modify (to cover up any lapses in the disguise from poor research) than a replicated outfit. In an emergency holoemitters could also be used as a crude cloaking device, like a Federation anthropological "duck blind".
The Star Trek Book describes the Mobile Emitter as being a piece of technology, from the 29th Century ship, that Starling appropriated. This confirms that he found it rather than having built it himself.
Starling equipped the EMH with a stolen autonomous self-sustaining mobile holoemitter, technology unknown to the 24th century.
Since it's compatible with the Voyager EMH, it seems likely that the ship had its own EMH.
In the “restored” timeline, surely Captain Braxton was aware of the mobile emitter that Voyager now had, even if it is paradoxical in nature - maybe it is this event that leads to the development of mobile holographic emitter technology - chicken or egg?
And while we’re at it, Braxton would only command the Aeon if he had crew to whom he gave orders. Single seater fighter pilots who have the rank of captain do not command their craft, they pilot it. For Braxton to be a command Captain, it follows he must have some crew, who could just be holograms he could use in extraordinary circumstances.
The time ship would likely need some method of disguising its pilot when they deep dive into the time stream. A relevant example from outside Star Trek is the Chameleon Jumpsuit from The Journeyman Project 3 wherein the player must use holographic disguises obtained from scans of locals in the time period to which the temporal agent is travelling. Since 29th century Federation officers are likely to have FAAAAAR more advanced technology, they would most likely have the ability to use a small portable holographic disguise without need for scanning locals. As such, the device would only be needed when the pilot is actively engaged in time missions so it would be stored on the time ship rather than worn as part of the uniform, much like away teams add a phaser when going to unknown planets but generally do not carry them while on duty.