Ethylene definitely has a role in the ripening of fruit.
Ethylene and fruit ripening is a scientific paper that discusses the chemical mechanisms of ripening fruit. From the paper:
The ripening of fleshy fruits corresponds to a series of biochemical, physiological and structural changes that make the fruit attractive to the consumer. Although these processes vary from one type of fruit to the next, fruits can be divided into two broad groups, known as climacteric and non-climacteric (Biale 1964). Categorization into one group or the other depends on whether or not a fruit exhibits a peak in respiration (McMurchie et al. 1972) and ethylene production during ripening. The sharp increase in climacteric ethylene production at the onset of ripening is considered as controlling the initiation of changes in colour aromas, texture, flavour, and other biochemical and physiological attributes. In contrast, the ripening of non-climacteric fruits is geneerally considered to be an ethylene independent process and little is known of the regulatory mechanisms underlying the biochemical changes.
From this we learn that some fruits produce ethylene as they ripen, and this ethylene is a chemical signal that controls ripening. The claim that certain fruits ripen other fruits is possible if the ripening fruit transmits a significant amount of ethylene to the other fruit. The first fruit would have to emit a significant amount of ethylene gas, which would then have to be absorbed by the second fruit. The second fruit would also have to be climacteric.
the ripening period of certain fruits such as apples, involving increased metabolism and only possible while still on the tree.
This health and nutrition website lists apples and bananas as climacteric fruits. It also elaborates on the basic claim in a way that is consistent with Ethylene and Fruit Ripening:
Climacteric fruits — those that can ripen after being picked — produce much more ethylene than non-climacteric, which cannot ripen once removed from the plant. Some fruits, such as apples and bananas, produce even more ethylene gas than other climacteric fruits. If you find yourself impatiently waiting for a climacteric fruit to ripen, you can speed up the process by placing the fruit in a brown paper bag to concentrate the ethylene, or even position it near an apple or banana.
Apple, banana, mango, papaya, pear, apricot, peach, plum, avocado, plantain, guava, nectarine, passion fruit, blueberry, cantaloupe
Citrus fruits such as grapefruit and lemon, berries such as raspberry, strawberry and cherry, grapes, pineapple, melon (including watermelon), pomegranate
After‐ripening of fruits adjusted by ethylene‐generating agents is a scientific paper that applied ethylene to kiwis and peaches. The ethylene speeded the ripening process in both. The fruit was exposed to 750 or 1700 ppm of ethylene gas.
Measurement of ethylene production during banana ripening found that ethylene production from bananas fell in the range of 2-50 µL ethylene/kg fresh material/h. If I assume a 2L container, 5 average weight bananas (~1 kg) producing the mid range of ethylene gas (25µL/kg/h) could produce 750 ppm in 30 hours. This is a comparable to the amount of time it would take for things to ripen on their own, but it is quite possible that lower concentrations of ethylene still have a ripening effect.
This article contains the following paragraph, which suggests that real scientists believe something similar to the claim.
Principle Bananas are climacteric. This method is designed to initiate fruit ripening with an ethylene-like gas such as acetylene or propylene and to measure fruit ethylene production by gas chromatography 1.
The article also cautioned that excess CO2 buildup in a container will inhibit further ripening of the fruit. If you put your fruit in too small of a container or a container with too poor ventilation, it may slow ripening.
Conclusion: Climacteric fruits produce and require ethylene for ripening. More ethylene will make them ripen faster.