READ CAREFULLY before further testing:
I missed the MKR1000 3v in my first reading.
That will complicate things as david said since many if not most do have pullups on the backpack.
I have used 5v backpacks and LCDs with 3v Aduino type devices. I have done this with ESP8266 devices.
It requires playing some games with the supply voltages and the pullups.
If the backpack has pullups on it (which yours does) then it is possible that it has blown the pins on the 3v device if they are not 5v tolerant.
The backpack uses 4.7k pullups which limits the input current so hopefully this isn't the case, but it is possible.
When doing mixed voltage i2c, the simplest way is to use a voltage converter.
But if you use a converter, you have to make sure to use the proper type or modify it for your use case.
Many are meant to interface 5v devices to 3v devices so they have a 3v voltage regulator on the low side.
In your case it is reversed in that the master has its own 3v power and you do not need or want this voltage regulator since it can cause a "short" with the 3v power already on your 3v Arduino.
To use the backpack at 5v without a voltage converter you have to play some games with the wiring
and voltages.
It is isn't difficult but it isn't as simple as directly using the 4 pin header.
In some cases you can get away running everything at 3v. That is the simplest case since you simply
hook the 3v supply to the vcc line on the 4 pin header instead of using 5 v.
That doesn't always work, and the backlight is often dim.
In some cases the backlight will draw more current than the 3v supply on the Arduino board can supply so be very careful if you go this route.
In most cases you have do as david mentioned and use pullups to 3v while running the backpack and LCD at 5v.
This is not difficult if the backpack does not have pullups on it as you simply
hook up the 4 pins (vcc to 5v, gnd, sda, and scl) and then add your pullups from sda and scl to 3v)
However, the photo of the backpack you have shown has pullups on it so you can't do that.
You will have to remove the i2c pullups on the backpack.
The pullups are on the left side of the backpack.
You will see a row of parts with: led, and 3 resistors labled 102, 472, 472
The pullups are the two resistors labled 472.
Those resistors are wired from the sda and scl pins to the vcc signal on the backpack.
If using 5v vcc on the backpack 4 pin header they will go to 5v and if interfacing to 3v they need to go to 3v.
Using a pullup to a voltage that is higher than the devices voltage (5v would be higher than the mkr1000 3v) can potentially damage the part if the pins are not 5v tolerant. (which it sounds like the pins on the mkr1000 are not)
In this case there is is a current limiting resistor of 4.7k
so the worst case current is about 1ma (I=V/R or 5/4700) = 1.06ma
That seems to be within the 7ma pin current so hopefully it is ok.
I have never understood why the arduino boys make boards with 3v only processors
especially since arduino is geared at less technical folks and "tinkering" is encouraged.
Both of which can create a situation of accidentally blowing up a board.
--- bill