Electronic – Temperature sensing with ADS1115

You're confusing the operating specifications of the chip with the readout format.

Basically, the Electrical characteristics specify the range of temperatures at which the IC is guaranteed to work properly. This is a physical characteristic of the device. As the datasheet says Operating temperature range, e.g. this is the temperature range over which the IC is supposed to work properly. It has no relation to the theoretical range of the register containing the measurement value.

The temperature sensor readout is specified as "-1 °C/digit". Therefore, each increment of the temperature output equals one degree C. My guess would be that the value of the temperature readout would probably directly correspond to the actual IC die temperature in degrees C, though you should verify this.

The two specifications are unrelated. It's like a car speedometer. My car's speedometer goes up to ~150 miles/hour. However, it's not intended to actually be driven at those speeds. As such, the readout range (0-150 Mph) significantly exceeds the intended operating conditions (~0 - ~85 Mph), as they're different characteristics.

Is there such a thing as an analog temperature sensor with 100mv per degree precision (...)

I think there are sensors like this on the market, but probably sold as industrial devices (not integrated circuits) and they may be a lot more expensive than LM34 integrated circuit.

or is there something I can stick on the end of the sensor to amplify its voltage x10?

Sure.

You can use operational amplifier in non-inverting configuration to amplify your voltage by 10. It's very simple to build.

Image from Wikipedia:

$$ V_{\text{out}} = \left(1 + \frac{ R_{\text{2}} }{ R_{\text{1}} } \right) V_{\text{in}}\!\ $$

You can use 2k\$\Omega\$ (R1) and 18k\$\Omega\$ (R2) resistors to get gain x10. Use 1% tolerance resistors if your measurement acurracy is important.

LM34 with x10 amplifier connected to output is good idea.

Probably (it depends on voltage range you need) you can use LM358 amplifier.

Edited:

It's low cost rail-to-rail amplifier that can work at input and output voltage = 0V, perfect for applications like this. Not all operational amplifiers can do that.