For example, if we have a film with ISO/ASA speed of 100, is it equivalent to setting ISO = 100 on DSLRs?
Sort of... because ISO values for film don't even have a 1:1 correspondence. For example, Fujichrome Velvia 50 is rated at ISO 50, but if you set your camera on 50, you will generally underexpose. Most film photographers I knew set ISO 40 instead. However Kodak E-100 typically did expose properly at ISO 100.
A digital camera set at ISO 100 will typically respond "as if" it were film that responded with a proper exposure at ISO 100.
Film photographers grew to understand the characteristics of the particular film they were using and adjusted accordingly. Digital photographers don't typically have that problem.
They try :) but is different in each camera. Sometimes it is close but it can be different.
Do you know the site Dpreview? Every time they review camera they say correspondence between real ISO and camera ISO. Example: http://www.dpreview.com/reviews/olympusem5/16 says this one slower by 1/3 EV.
In theory, yes. ISO is short for the ISO 5800:1987 standard which defines what sensitivity means and how it is measured. That standard is intended for [negative] film but digital cameras makers have been using it to measure sensitivity too. There are however room for interpretation when you consider things like RAW conversion, color-spaces, etc, there are not always exact matches.
As @Zak said, some sites measure this and you can actually get a very price chart for each RAW-capable camera at DxOMark. They only measure RAW output to avoid artifacts introduced by JPEG conversion. For example, for the Canon 5D Mark III (Select the Measurement then the ISO Sensitivity tab to see the chart), ISO 100 to 102400 are extremely close to the official measure. ISO 50 which is an expanded sensitivity is not.
Recently I've looked into this. For decades I worked as a professional, outdoors and indoors, using studio flash equipment, as well as a lot of available light photography. Within the 120 roll film the sheet had a guide setting, which was actually highly accurate. Example 125 ASA bright sunlight said 1/125 combined with f/11 or f/16. It didn't matter where you were, in Egypt or Arizona. Frankly, I hardly used a light meter, except for in very difficult situations. If I would set a present DSLR on the same settings, the result will be about 1 to 2 stops under exposure, depending on which DSLR.
So in my opinion, the answer is are they equivalent; No!
Another thing about the so-called high tech meter systems within the DSLR; none of these work. In fact the most approachable right method of measuring is to measure the direct light, which should always give you Neutral Density. The given measurement within any camera often misguides. Because of the simple reason, as soon as there's a person in front of the camera with a white shirt is taking 50% of the screen, you will have to correct this. Simply said; the build in TTL is an aid device, but far from perfect.
Actually, no. The whole idea of calling it "ISOxxx" in digital cameras is fundamentally flawed as the ISO standard pertains exclusively to photographic film.
The proper term for digital cameras is EI, or Exposure Index, which is an attempt to match digital sensor photo sensitivity to ISO sensitivities for film base. As stated in other answers, there is a spread for film base sensitivities, this being replicated in there being a spread in sensor sensitivities as well (and potentially a greater spread) as different sensor makers use different tables.