Can there be sunlight in Australia and Texas at the same time?

Yes.

Let's go with your assumption that the call took place in the middle of September.

Time and date lists sunset time as 19:33 on September 14th in Dallas (I couldn't find Mircale, TX), and sunrise time as 6:20 in Melbourne on the 15th.

With the 15-hour difference, if the call took place at 8:30 on September 15th in Melbourne, it would be 17:30, September 14th in Texas and both locations would be roughly two hours since/from sunrise/set and it would be bright outside at both locations.

I don't know what day the scene takes place, but assuming it is today, 8th July 2020, then Wolfram|Alpha1 calculates that the sun rose in Melbourne 4 hours before it set in Austin. The additional day is due to the cities being on opposite sides of the international date line. For 8 Jan there is 4.5 hours of shared sunlight.

At 09:52:52AEST 9 July 2020, (18:52:52CDT 8 July 2020) 2 the sun will be just under 20deg above the horizon in both cities.

Others have answered that it is, at the very least, possible for it to be daytime in both places but impossible to be midday in both places. I'd like to answer a related question, "how close to solar midday can both these locations get, and when?"

As for "how high," that is easily found by taking half of the great-circle distance between the two places and converting it to degrees on the Earth's surface. Using Austin as a stand-in for the uncertain location of Miracle, we find that to be 64.3 degrees from noon, or 25.7 degrees above the horizon in both locations.

As for when, that's a little bit trickier. The midpoint of the shortest great-circle path between Melbourne and Austin lies at (-7.200, -152.217) in the central Pacific ocean (obtained via Google Earth). The answer to this question is, "When will the subsolar point come closest to this location?"

Using timeanddate.com, and some old-fashioned trial and error, I have found the closest time and date (in UTC) to that time for 2020, to the minute: 10 October at 21:56 UTC, which corresponds to 11 October, 8:56 AM in Melbourne, and 10 October, 4:56 PM in Austin. At this time, the sun is 25.7 degrees above the horizon in both places.

The exact date will depend on the latitude of Miracle (From 25 September at the point in Texas furthest North to 21 October at the point furthest south), while the exact time will depend more strongly on its longitude (from 21:46 at the point furthest East to 22:20 at the point furthest west, but still varying with latitude).

EDIT: It occurs to me that there are two times of year where this takes place. The other time of year, assuming again Austin, is 1 March, 22:21 UTC, which is 9:21 AM in Melbourne and 4:21 PM in Texas. Also, I corrected 7:56 to 8:56.

Comparing times is not a great way of checking for daylight as time zones are very unevenly distributed.

solar "noon" can occur as late as 15:00 in western portions of China

Better to just some simple geometry, logic, and, as pointed out in the comics, some knowledge about Earth's movements relative to the Sun:

If

1. half the globe is in daylight at all times;
2. the furthest away two points can be is halfway across the globe;
3. and the Earth rotates fast enough relative to its revolutions around the Sun and the apparent size of the Sun in the sky such that all points on the globe can view the Sun at some point during an equinox;

then there must be some time when any two points are both in daylight.

Alternatively, just look at a globe and try to see both Texas and Australia at the same time:

Note that while they can both be in daylight at the same time, they cannot both be near solar noon at the same time. So that would be an inaccuracy in the show.

On a Summer’s day like today, 08 July 2020, the sun will rise in Texas at about 6:30 AM and set about 8:30 PM Central Daylight Time. The sky will be light around 30 minutes before sunrise and about 30 minutes after sunset. That is roughly 15 hours of daylight. Theoretically, first light and last light may happen simultaneously in the two locations during the Northern hemisphere summer months. Unfortunately, the US Navy sunrise/set calculator is currently down for service for me to make a more scientific analysis.

It's possible.

Google says 8am in Melbourne is 5pm in Austin

It's technically on a different day, which is possibly where the confusion lies, but around September both would be somewhere near equinox, so both should be in daylight around that time.
15 hours one way is 24-15=9 hours the other way, crossing the International date line.