Yes, they’re the same sufferings of your life, no matter what. When you consciously join your sufferings with those of Jesus you copy what St. Paul did when he wrote: " I Paul am made a minister. Who now rejoice in my sufferings for you and fill up those things that are wanting of the sufferings of Christ, in my flesh, for His body, which is the church"
Yes, however, you can offer up your suffering in general or specific sufferings: a paper cut, a bee sting, the tiny sting of humiliation when someone at work corrects you, the burn on your tongue when you take that first sip of coffee too fast in the morning.
We are asked to ‘put on Christ’, that is, to be like Him. He asks us to pick up our cross and follow Him. Offering up our suffering is a way to do just that, pick up our daily crosses and not just drag them around after us.
We are also asked to pray always. Offering up our daily sufferings is another way to do that, as well. Always turning to God when ‘bad’ things happen and making the best of difficult situations. How better to make the best of a paper cut than to offer that bit of pain for the Holy Souls in Purgatory?