So I'm an electrical engineer (actually computer design), although not a "qualified" one for any of this
Basically, an EVSE is:
1) A double pole relay between the wall power and the big terminals of the car connector. Relays contacts have both voltage and current ratings. Current will be the same, and while it's likely the relay is rated for 240V, it might not be. You'd have to tear apart each cord to be sure (if the relay is labeled), because even some have 240V rated relays there's no guarantee the manufacturing run wasn't optimized before or after that.
2) Low voltage circuitry controlling the relay coil, using the small terminals of the car connector for handshaking (this handshaking is what tells the car what current can be drawn). This part shouldn't be affected by the charging voltage (because of #3)
3) A power supply to provide the low voltage for #2. Just like most phone chargers etc automatically handle 120-240V inputs and step down to the same fixed voltage, as in #1 there's no guarantee unless the part in the particular cord is labeled and rated for 240V.
In electronics manufacturing, there's often a single design that is tested during manufacturing, and due to process variation some parts meet spec A and other parts only meet spec B. For example, switching power supply chips that don't meet 240V specs but do meet 120V specs might be binned for use in 120V-only cords. Depending on volumes needed, some 120V cords might have chips that could have met 240V, some might have 120V-only chips. And because it could be on the margin, it might work sometimes on 240V but fail when the outside temperature is a bit higher, or lower, or the house voltage is a little higher or lower. Best case, the out-of-spec voltage just causes the EVSE to stop working, worst case (unlikely but possible) it starts a fire.
I don't mean to be a wet blanket here -- we make risk tradeoffs every day in life, and it's not unreasonable to decide the risk here is low enough to be worth doing anyway, but people should just be aware of the risks. I'd be comfortable doing this myself in a place where I can keep an eye on it, but I probably wouldn't want to leave it unattended, especially in a garage underneath my kids' bedrooms while they slept, for example.