Article II of the US Constitution currently prohibits an elector from voting for two people from his home state.
The electors shall meet in their respective states, and vote by ballot for two persons, of whom one at least shall not be an inhabitant of the same state with themselves.
Some background from History.com:
Under the original system, electors did not distinguish between candidates for the nation’s top two offices; the candidate with the most votes became president, while the runner-up became vice president. The 12th Amendment, adopted in 1804 after two chaotic elections, mandated that electors cast separate ballots for president and vice president. However, the rule preventing an elector from voting for two people from his home state remained in effect under the new system.
And from PolitiFact:
The idea behind the restriction was to dilute the power of the big states and to encourage electors to look beyond their political allies.
However, this does not mean that the President and the Vice President cannot be from the same state. This rule would only matter if the election was exceptionally close and came down to one state (the 2000 election was almost affected by this rule as Texas was the home state of both Bush and Cheney; Cheney later registered to vote in Wyoming instead, side-stepping this issue).
Let's use the Bush/Rubio hypothetical. Bush could choose Rubio to be his running mate. And if the two won, Florida's 29 electors could vote to make Bush president, but not to make Rubio vice president (or vice versa). If Bush and Rubio started with 299 electoral votes or more, that wouldn't matter and Rubio would still be vice president. But if the election were close, things would get complicated.
Also, it's worth noting that "electors" in this constitutional amendment refers to the voters (of all 50 states and D.C.), not members of the electoral college (which would have been abolished if this is passed). From the joint resolution linked in the question:
Each elector shall cast a single vote for two persons who have consented to the joining of their names as candidates for President and Vice President.
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The pair of candidates having the greatest number of votes for President and Vice President shall be elected.