Historically, based on geographical varieties, plants gew as they could adjusted themsevies with the local climates. On the other hand, the need of human for various nutricians to survive, and limitation of local sources motivated them to seek, transport new plants as a business, based on their customers' need, on the way to back- home (British's spice- business from china to Europe). In addition, well- known routes of Silk Road or else, were an effective variable on migration of plants from East Asia to Middle East,North- Africa to West. Expectedly, the global world of current era talks for itself regarding the rareness of being a native, no matter in what.  

The plant must have evolved in one of the two places and then spread to the other. Either that or it is two different species that are difficult to distinguish from each other. Genetic analysis should be able to determine the spread pattern and probable place of origin. 

There are lots of circum-boreal examples (not every interesting, given the small geographic barriers).  Also several (spore-dispersing) ferns. Two of my other favorites are Themeda triandra, native to both African and Australia, and the three-continent-hopping Arabis alpina.

It is possible just as the above researchers have alluded to. If the two areas you are looking at have very similar climatic and environmental conditions. You may have to consider looking at population genetics (Phylogenetics in particular) to assess the relationship if any.

we have many examples of plants native in several countries and continents, especially the aggressive species, once cultivated in different country or continent, after a short time may become native like Oxalis sp., Ailanthus, Acacia ect.

As a few others have said, there are examples of grass and fern species (e.g., Pteridium aquilinum) found on more than one continent as examples of natural long distance dispersal. There are also many mosses that are distributed across more than one continent, e.g. Polytrichum juniperinum is reported to occur on every continent with the exception of Antarctica.

I would consider the possibility that in one of the places the plant may be cryptogenic. Lots of species have been transported, either intentionally or accidentally, by ships and early voyagers, and, unless it has been proved that the species has the capacity of very long dispersal, I would incline for an early introduction to one of the sites.

I know a good example from the Atlantic Islands: Cakile edentula, found in some islands of the Azores archipelago (Sao Miguel and Faial) and present in North America. Another good example is Ipomoea pes-caprae, a pan-tropical species present naturally in the sandy coasts of Africa, the Indian Ocean, South America, etc. There are more examples of halophytic plants with a wide distribution around the world.

Hi again.

One of the best examples I know, apart from several halophytic plants, is Erica arborea. It's present (naturally) in southern Europe, some Macaronesian archipelagos and several parts of Africa and western Asia. In any case, populations from Madeira and the Canaries have been described very recently as Erica canariensis. 

Many thanks to everyone!

Hi again.

A further example of a plant with a wide distribution in different continents is Sesuvium portulacastrum, common halophytic plant in the Cape Verde Islands, the African continent, Asia, North and South America and Australia. 

Best regards. 

Coconut (Cocos nucifera) is widespread in tropical areas.

Common juniper (Juniperus communis) is found in Asia, Europe, Africa and North America, including Iceland, Greenland and Japan. As far as I know, no other tree/shrub species has such a natural range.

https://en.wikipedia.org/wiki/Juniperus_communis#/media/File:Juniperus_communis_range_map.png

yes is possible. For example carob tree this ancient tree is native around Mediterranean countries/Europe, Africa and Asia/.