Is there a way to see if the picture of the channel I am currently watching is in 720p or 1080p and do I have any control over that?

All DIRECTV STREAM content is delivered in 720p and 1080i, there is no other way to adjust it. If your wife thinks the picture quality is poor, that may be due to your internet connection. 

Shows that are from 2005 would be likely upscaled by DIRECTV or the content provider to 720p/1080i

Select sport events are upscaled by DIRECTV to 4K.

Keep the "-" button on the Osprey (DIRECTV STREAM box) remote pressed for 5 seconds, and a pop-up will open and show the details for the channel you're watching. 

For the Dtv Stream box---go into settings--system--display and turn off HDR and set screen resolution to 1080p. Makes a big difference.

Thanks, everyone, and I will try what you suggested Ludwick577.  I will report back.  

OK....my internet service speed is 300Mbps, and the actual speed is right at 370.  It's AT&T fiber.   GTyz, thanks for the tip on how to see the screen resolution, and Ludwick577, thanks for the instructions on changing the resolution.  When I went to change the resolution, I found that my HDR setting was already off, but the resolution was on the automatic setting where it can go up to 4000p, so I did change that to 1080p.  Note also that this is a brand new 65" Sony XR65X90CK television, so it's a medium to upper medium-grade TV.

I just have the Entertainment package, which is the entry-level package with, I think, 75 channels.  Following GTyz's instructions and after making the changes recommended by Ludwick577, I looked at the resolution of all 75 channels.  I started with USA TV, where I was watching an old Law and Order episode. It actually looked very good, and I checked the resolution and it happened to be 1080p.....hold that thought.  I then went through all the other channels in my plan, and all but 2 were 720p.  Some of them looked better than others at that same resolution.  Then I made my way back around to USA TV and it did not look quite as good as when I started this process. I checked the resolution of the USA channel and it was now 720p.  

This leads me to a few questions:

1.  Who/what determines the resolution of a streaming channel?  Is it AT&T or the channel owner?  

2.  Does the speed capacity of my internet service affect whether a channel is coming through at 1080p vs 720p?  According to DirecTV's recommendations, my 300Mbps should be plenty fast enough for our streaming.....just 2 old folks who watch one TV at at time

3. Is the resolution of a given channel variable?  Will it sometimes be 720p and sometimes 1080p?  What causes that?

4.  Will the resolutions of all DirecTV Stream channels eventually be at least 1080p? Is that a matter of AT&T making the necessary equipment and infrastructure investments to achieve that?  I don't have an internet provider choice.  I must use AT&T because no other provider will serve my house. 

Thanks everyone.

Hold the - button on the stream remote. 
DTV stream channels are 720p or 1080p a few 480.

Networks decide what resolution they broadcast in. Directv Stream just carries what the networks/channel provide.

Also, you said HDR on your Dtv Stream box was already off. Make sure your Sony Tv's HDR is also off on the HDMI input your Dtv Stream box is connected to.

Also, I use Rtings to calibrate my tv's. Here is how to calibrate your tv for optimal picture 

https://www.rtings.com/tv/reviews/sony/x90k/settings

I've got it adjusted pretty well now.  Thanks, everyone for your help!

Answers:

1. As noted, the content provider/channel.  As far as we can tell, DTV just acts as a pipe.
2. No.  It won't come through as 1080p.  It'll be 1080i, 720p, or some SD resolution, depending on what the source provides.  However, you can force some streaming devices to upscale (or perhaps in some cases downscale).
3. No.  But some channels that show reruns may show those in the resolution that they received them in rather than rescaling them, so conceivably the resolutions could change program by program.  I believe the major networks would upscale all older content, however.
4. That is up to the content providers.  As I said, right now DTV doesn't appear to rescale anything.  We don't know if DTV's current systems could send 1080p if a content provider sent that to them, but I'd assume that if/when that happens, they will have to deal with it.  With the spread of ATSC 3.0 (the new broadcast standard) it seems likely that at least some content in 1080p will start happening fairly soon.  (And BTW - DTV is a separate company from AT&T and has been for about a year and a half.  The fact that your internet provider is AT&T is irrelevant to the DirecTV Stream service.  And vice versa - you being restricted to AT&T as an ISP doesn't mean you are restricted to DirecTV Stream as your streaming TV provider - you could get any of them.)

WRT your question about 2005 era shows - the first HD broadcasts started in 1998 I believe, so some HD content became available then.  (Anything older would likely be in SD, though a few older syndicated shows have apparently been rescanned into HD.)  I believe all the major broadcast network content would have been HD well before 2005.  However, the "cable channels" were all over the place - some switching to HD before 2005, some much later, and their content would likely have tracked with that.  There is a list (of not known accuracy) here of when many of the cable channels switched.  Of course, they often show older content, and they might or might not have upscaled that older (SD) content after they switched.

Jrandomuser, 

Thank you very much for your great explanation 

Ludwick response “For the Dtv Stream box---go into settings--system--display and turn off HDR and set screen resolution to 1080p. Makes a big difference.”

This did help. I was wondering why when watching golf on Peacock it was sharper than watching it with DTV stream. I had HDR on and that was my problem.