It varies by state from what I recall. In Texas they do have serial numbers:
Ballot Numbering Requirements: The Texas Election Code establishes the following ballot numbering requirements in order to track and account for the total number of ballots used in an election, how they were used, in what locations, and to account for the ballot stock that was unused Any ballots used in a Texas election must be (1) numbered consecutively, beginning with the number “1”, (2) numbered so that a specific range can be linked to a specific polling place, and (3) must be distributed to voters non-sequentially in order to preserve ballot secrecy. [...]
For electronic votes, the Texas numbering system is a bit different:
Ballot Tracking Through the Voting System Software: The procedures below have been approved by the Secretary of State’s office as a means of fulfilling ballot numbering requirements through the system software and/or the use of additional peripheral devices. Any vendor seeking to utilize a software solution for ballot numbering must have their solution approved by the Secretary of State.
ES&S ExpressVote – Tracking Ballot Numbers Through the ExpressVote Activation Card Printer/ExpressLink Software
Each ExpressVote Activation card printer that is assigned to a specific polling place will be given a two to three digit alpha code. This code will be printed on each Ballot Card as it is generated for the voter.
In addition to the alpha code, each polling place will print a randomly generated serial number between 1 to 99,999 on the card along with the identifying information related to the election.
The system will generate a report showing which ballots (based on their serial number) were used at each location. This report must be retained with your precinct election records. [...]
Now I don't exactly what the Trump representative is talking about in that video, but they seem to talk of something specific to Dominion machines. A number of claims related to these have been debunked by the media, however there doesn't seem to be a clear or specific-enough claim in that section of the video relating to these (serial?) numbers. In fact, I'm very inclined to agree with Giter's comment on that:
just listened to what she was saying around what you quoted, and it sounds like she's saying that two different states recorded the same six digit number of votes: "We have evidence of different numbers of votes being injected into the system, the same identical six digit number multiple times in at least two states that we've analyzed so far, and I'm talking like 341,542 votes for Biden, and 100,012 for Trump". In other words, I think the claim is that two states reported the same vote totals at some point, not anything about unique ballot IDs. – Giter
i.e. this is just some variation on the claim that the Dominion machines fabricated votes etc.