Each senator is in one of three classes, and at each election all the senators in one class are elected. In the 2018 race, for example the "class 1" senators were elected. That regular elections for different classes happen at different dates is not merely a convention, it is mandated by Article one of the Constitution.
In addition to the regular elections that choose senators for a period of six years, and are held every two years (as mandated by Article 1, as modified by Amendment 17), there are also "special elections". These occur when a senator dies, resigns or is otherwise removed from office. Special elections can be held at any time. The senator elected in a special election serves only for the remainder of the term. So if senator dies two years into their 6 year term and is replaced in a special election, the new senator may serve for the remaining four years before having to face re-electionn
It is possible for special elections for a Senate seat to be scheduled on the same day as regular elections. In which case a single ballot paper can be used for the two Senate seats (and the House seat, and any other elections occurring at the same time) This saves money and is convenient.
For example in 2018, in Minnesota, Tina Smith was elected to fill the class 2 Senate seat in a special election, while, at the same time, Amy Klobuchar was elected to the class 1 seat. Klobuchar will now serve for 6 years, while Smith will face re-election in two years time in the class two senate elections.
The constitution is allows states to organise the election as they see fit. It only requires that:
When vacancies happen in the representation of any State in the Senate, the executive authority of such State shall issue writs of election to fill such vacancies.
This would seem to require that a separate election is held for the vacant seat. At any rate, there is little to gain in holding a joint election. It creates a confusing system in which the parties would game the system, entering 1 or 2 candidates according to tactics and reducing the ability of the electorate to choose their senators. No state has done this.