Is a swallow and a martin the same bird?

Yes and no.

They are both in the family Hirundinidae. Within the Hirundiniae, the name "martin" tends to be used for the squarer-tailed species, and the name "swallow" for the more fork-tailed species; however, there is no scientific distinction between these two groups.

Of the same subfamily.

The swallows and martins are a group of passerine birds in the family Hirundinidae which are characterised by their adaptation to aerial feeding. Swallow is used colloquially in Europe as a synonym for the Barn Swallow.

This family comprises two subfamilies: Pseudochelidoninae (the river martins of the genus Pseudochelidon) and Hirundininae (all other swallows and martins). Within the Hirundiniae, the name "martin" tends to be used for the squarer-tailed species, and the name "swallow" for the more fork-tailed species; however, there is no scientific distinction between these two groups.[1] The family contains around 83 species in 19 genera.

https://en.wikipedia.org/wiki/Swallow

Swallow is the name of the only swallow normally found in the UK. This name is not used in Europe, where each language has its own name - Schwalbe, zwaluw, hirondelle, svala and so on. It's interesting to note that the European languages I've referred to make no martin-swallow distinction. They're all swallows. (ref: Peterson, Mountfort and Hollom: Field Guide to the Birds of Britain and Europe, Collins.)

Baloo: There's also another swallow found in Britain, the House Martin.

https://en.wikipedia.org/wiki/House_Martin

We have Sand Martins as well. My point is that they're not called swallows in the UK, but martins. We distinguish martins from swallows, unlike your country and probably the rest of Europe who class them all as 'swallows'. Don't ask me why. Had too long a day. Basically, in English speaking countries, swallows and martins are not the same, but in foreign speaking countries they are. They are all hirundines, however, wherever you are. Like herons and bitterns, and swans, geese and ducks: scientifically one group, but they are easily distinguished (mostly) by non-scientific people into their popular constituents.