I was looking at my vaccine card to see if there was a reference number or bar code or something
There is not. It does show what batch of vaccine I took. I looked at the paperwork from getting vaccinated and there was a QR code there but I don’t see any way for someone to check your vaccine record easily. Seems like they should have put something on the card. So if they come up with some kind of vaccine passport it’ll probably be a big effort, though they ought to have all the info already. Really they have enough info to just mail something like that out without a person having to initiate it, but I’m sure there all kinds of issues with that.
I’ve read about fake card being sold which I am sure will end up being a big problem. I’ve read of people getting busted for having fake cards. That probably took some effort, have to track an individual. If there were some reference number in the card they ought to be able to have an online check which would not show names, but there isn’t.
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It means that all of the product with the same lot number were manufactured from the same batch of ingredients mixed in the same gigantic vat, and any vials filled with that product will have the same number on their labeling. That way they can be traced back to the date the batch was made and where (and in some cases, what time, and who QA'd it, etc).
When they change batches, they change lot numbers (or are supposed to... some of the schlock places will mix them and distribute/sell them anyway 
).
I know a few months ago I had seen a link to check the lot numbers for expiration dates from websites that were tracking them, and found these for example -
A Moderna one - https://www.modernatx.com/covid19vaccine-eua/providers/vial-lookup
A Janssen (J&J) one - https://vaxcheck.jnj/
Pfizer - haven't been able to find a lookup tool (and neither did the Philly Health Department)!
Some other info on the various related codes - https://www.ama-assn.org/find-covid-19-vaccine-codes
Obviously as long as you got it before July (which I know you did), you're fine! 
I had checked mine back in May (and saved screenshots) and the 1st shot's lot expires Sept. 8, 2021 while the 2nd one expires Oct. 2, 2021. I had received the 1st dose on 3/17/21 and the 2nd on 4/21/21. So it was a 6 month expiration for each.
I had really lucked out getting those appointments because they popped up when I was fruitlessly searching for vaccine appointments (3 months after signing up with the city and hearing nothing) and found a little pharmacy not far from me that showed up on a list of smaller pharmacies recently approved to give them that would start doing them "in the future", and that pharmacy allowed people to put themselves on their waiting list.
About 5 days after I added myself to their list, they sent me an email late the night before opening up appointments for the first time, offering me to pick an appointment time, I saw it early the next morning, and jumped right on it to schedule my first appointment that same day! 
My first shot (got it on February 26th) expires on August 17, 2021 (my mother's birthday) while my second shot already expired on April 28th! Fortunately I got my second shot on March 28th. BTW, I got the Moderna at the CVS in Saratoga, NY.
Back then. I also passed on the link to your post to my friend who made the appointment for me. He'd been trying for days, like we all were back then, and when he managed to get an appointment when they started offering it at CVS, I happened to be in the dentist's chair in Albany (not something I do all that often), so by the time he got hold of me, the local CVS was booked - so I had to go to Saratoga. Not that I minded, I was grateful and relieved! 
BTW, he's also on DU, so you may get a reply from him, too. 
What is ironic, is that a couple days before I was due for my scheduled 2nd shot, the city finally contacted me to make an appointment (literally almost 3 months after signing up on their list). I then got emails, texts, and phone calls to my land line to sign up - and it was all for the mass vaccination site down at the PA Convention Center, which is inconvenient for me due to parking issues and then having to stand outside in the cold, etc.
I literally had a Riteaid pharmacy near me that was offering them but initially they f-ed up and their system had a supposed "bug" that was rejecting city zip codes (despite that store being located in the city) and signed up only suburban residents (who would be using the city's allotment instead of their own county's allotment). And then when that was exposed in the local media and they supposedly fixed it, now due to the "phases" and despite my having eligible underlying conditions, I was still rejected from making an appointment there (under 65, not in a designated occupation since I was retired 
, in the wrong zip code because it was already one of the higher vaccination areas). Just one excuse after another. 
It was just an outright mess.
When I got my first shot, they made my second appointment in (approximately) a month at the same CVS. Other than having to drive almost a half hour through a rainstorm, I had no issues. There was no wait - other than the 15 minutes afterwards - and I was in and out in about 20 minutes. I felt pretty miserable later that night and the next day - really bad chills and I was afraid to get up since I felt like I might pass out - but I knew that there might be side effects and I was so grateful to have gotten it, I figured that was a small price to pay.
Pfizer's maximum batch size is 3 million doses. That's about 660# of active agent. In the process solution, it would be about 800 gallons, before concentration.
In my former world, "giant vat" doesn't start until 20,000 gallons.
And, nobody has used "vats" since the 70s. Especially in pharma!
I just had to! 
And I call them "vats" and have seen them up close and personal (although not vaccine stuff).
...in the first Batman movie???
Now, those were some big vats!!! 
That was a small vat.
There are bigger ones, but you don't wanna know what's in them.
that was mentioned.
They barcode them too and have barcodes on the trays and cases so they can track them all from the plant, through the transport to/from distribution points, right to the destination holding facilities.
that this type of idiocy was happening all around the nation. The politicization and ignorance can be breathtaking at times, and perhaps not so much a dig at the lowest level administrative staff who are "following orders", but to their bosses, who are on a sabotage mission.
It has the dates, dosages, lot numbers, and location of my vaccines — they are stickers that were affixed to the card. There’s spaces to enter two more records. I thought this was how all cards were. 
although I have seen cards from some sites (probably max-vaccination ones) that have all of that info on pre-printed stickers that they can peel off and put on the cards.
I just saw a local paper story that after sortof pooh-poohing taking pics and sharing vax card images online earlier this year, has now started recommending taking pics of it (front and back) as a "proof" (without a need to carry the actual card around), but still obviously not sharing it in case scammers got ahold of the image.
First thing I bought after I got my first vax appointment and shot, was a 6-pack of card holders (4" x 3" ones with lanyards although I didn't need the lanyard but the ziploc-tyle waterproof sleeves came with them). Gave the extras to my sisters for their families.
