jwh20 is right, especially about the electrician.
However, if your breaker is DESIGNED to allow connecting two feeds, you may be able to run a second line off the existing 30-Amp breaker to power the second AC unit (local codes may override NEC minimum requirements).
Another however ... you can check the ratings of your AC units; they may both be OK on one breaker if their combined load is less than 30 amps. They may cause lights on the same circuit to dim, especially when they start, and if they both kick on at the same time, they might overload the breaker.
Have your electrician consider both those options.
UPDATE: I’m too new to comment on answers, so I’m limited to editing this one, but my response is still relevant to this answer, too. Breakers do not provide capacity; they limit flow (“current”, “amps”) before overheating occurs (all electric flow generates heat; too much flow means too much heat and possible damage or fire).
GFI’s add a sensor that detects when there’s a “leak”, say when there’s an unintended connection via water in a device that’s not waterproof or maybe a frayed wire makes contact with something or maybe a nail is driven through a wire, but there’s not enough additional flow to trip the overload trigger. AFI’s monitor the waveforms for telltale patterns that indicate arcing. Both trip well before too much heat is generated, the condition that traditional breakers already protect against.
Your nameplate shows max current (flow) of roughly 5 amps; that should occur for short time as it kicks on. It then settles to about 3 amps. You could in theory run 5 identical units off your 30-amp breaker. (Heat does not disappear immediately, so running 6 would likely cause nuisance tripping).
Another breaker between would add almost nothing for protection, since the original breaker will always see as much or more current and likely trip first. With a dead short, the two would “race” to trip first. The exception would be if the short were at the end of the line and the mid-line breaker tripped first, your first unit would still have power.
If the mid-line breaker were smaller, it might be more likely to trip first for a fault. It would likely trip first on smaller overloads or if the load happened to be less than 30 but more than the smaller breaker, the 30 would likely not trip (all breakers can mis-operate)
Regardless, a second breaker serves too little purpose if you’re not wanting to make it a “sub panel” and run thinner wire by using a smaller breaker.
The other comments about cooling capacity and latent vs sensible heat are spot-on; you want the units to run “long enough” before cycling. So there’s a good chance you’ll be ok adding the second unit.
But let a knowledgeable, qualified person look at your entire situation! Use this and other answers and comments to make sure your pro isn’t less than you need.