Sashiko, Boro, and Medieval Mending.

Both these techniques, as far as I can gather, were born out of necessity and poverty. Mending clothes was something done everywhere and in every time period when the cost of the textile, or the garment, was high enough to make mending feel worthwhile. In some cases, mending was done to be as inconspicuous as possible, while in other cases, the higher priority was to fix it in some way so that it would be okay to use again, with optics playing a secondary or no part. Which kind of mending was done probably depended on many different factors, including the personality of the mender and the owner of the garment.

Mending things usually is associated with poverty - if you are rich, you can afford to have new, unworn and undamaged things, and thus no mending. While having mended stuff would probably have been the norm for most people during most of human history, this association did its part to lead to our modern culture of throw-away clothing, bought very cheaply and in poor quality, worn a few times and then ending up in a landfill, or getting burned in some wast combustion plant.

Today, mending is becoming more fashionable again, in a turn against wasteful consumer culture. This also means that visible mending is turning into a kind of fashion statement - including the aforementioned techniques of sashiko and boro.

So did we have these, or similar techniques, in the Middle Ages in Europe? Whether that is answered with a "yes" or a "no" partly depends on how you see the different aspects.

Firstly and most importantly, we did have a similar tradition in the widest sense - that garments and textiles were mended, re-used, and re-purposed; especially in poorer areas or by poorer people, who could not afford to buy shiny new things all the time. For the boro tradition, what comes to my mind first is the textile equipment of the Man from Bernuthsfeld, whose tunic was put together from many different patches of many different fabrics:
However, this tunic is a special case - there's nothing else made in a similar fashion from the Middle Ages, as far as we know, and there is a lot of different speculations on why the man wore something like this.

There is a tradition in the European Middle Ages, though, that could be seen as a parallel to Japanese boro: Re-using the fabric of a garment to make something new out of it. This was done with garments that had a high value, either a tangible one (a very costly fabric) or an intangible one (something worn by a saint). Examples of these re-purposings are two Swedish altar frontals made from expensive garments that were donated to the church, or the Sixtus chasuble in Germany which was pieced together from another fabric, or St Birgitta's mantle in Sweden, which was made from what was a tunic or dress before.

There's also plenty of evidence for patches being sewn onto garments, sometimes in several layers, like in the underpants from Lengberg. Mending was not restricted to lower-class garments or less expensive fabrics either; for instance, the golden gown of Queen Margareta also has mends and patches. Off the top of my head, though, I don't know of patches reinforced with running stitches, or these stitchings being used as decoration.

So - I'd say that we had some aspects to mending traditions in Europe that could be considered similar to boro in Japan, but nothing that exactly or closely parallels the two Japanese techniques.

 
 
Read more about:
The Bernuthsfeld man:
Farke, Heidemarie. "Der Männerkittel aus Bernuthsfeld. Beobachtungen während einer Restaurierung." In Textiles in European Archaeology. Report from the 6th NESAT Symposium, 7-11th May 1996 in Borås, edited by Lise Bender Jørgensen and Christina Rinaldo, 99-106. Göteborg, 1998.

St Birgitta's mantle:
Andersson, Aron, and Anne Marie Franzén. Birgittareliker inlånade till Historiska museets utställning "Birgitta och det Heliga landet" 30 november 1973-17 februari 1974. Vol. 59, Antikvariskt arkiv. Stockholm: Almqvist & Wiksell, 1975.

Margareta's Golden Gown:
Geijer, Agnes, Anne Marie Franzén, and Margareta Nockert. Drottning Margaretas gyllene kjortel i Uppsala domkyrka. Stockholm: Kungl. Vitterhets Historie och Antikvitets Akademien, 1994.