This could provoke a fevered search for “true identity,: I suppose. That’s not what I have in mind. I am thinking of being proud of “who I am—today” as opposed to “who I was—yesterday.”
To make this work right, we need two elements. We need a self of some sort. As everyone knows, there are legions of notions of how to say just what a “self” is.  All I will need to mean is “the person you have in mind when you say I.” And you need some kind of emotional palette which includes the emotion “pride.” 
That’s pretty much it.
The trick is not to identify I with a system of abilities you once had. It is common to refer to the self you used to be as “who I once was,” but I think it would be perfectly appropriate to refer to that previous version of the self as “him.” Or, of course, “her.”  This would enable a old man, struggling every day with the reduced capacities dementia has dealt him, to remember the self he was as CIO of a major company and say, “He really knew his stuff and he required competence from the whole staff.” Or, more briefly, he might say, “I am so proud of him,” referring to that CIO self, a role he had played himself. 
What do we gain from this little verbal shuffle? It saves I to refer to the person he is now. Being proud of this person is the task I would like to address. Referring to earlier versions as “he” or “she” will clarify just what the task is.
- Is it really possible to be proud of keeping your sense of humor when you keep forgetting just where in the conversation you are or what your daughter’s name is, the one who is visiting you at the moment? Yes It is. Does that require that you never forget your daughter’s name? No. It doesn’t.
- Is it really possible to be proud of going to some kind of physical exercise or physical therapy every day? Yes. It is. Does that require that you walk as far or stretch as expansively or swim as efficiently as you once did? No. It doesn’t.
- Is it possible to be proud of an essay you have written when it represents the best thought and the most adroit expression you are capable of?  Yes. It is.
- Is it possible to present yourself confidently—it’s the appearance of confidence I am referring to—to new people as if you thought it would be worth their while to get to know you? All the time you are doing that, you are aware that they might not find you as desirable as the image you are presenting, so there is a very real risk of being rejected. What you know about yourself is that you really hate to risk rejection, but that if you don’t run the risk from time to time, your world will get smaller and thinner. Is it possible to be proud of yourself for running that risk? Yes. It is. Does that require that you succeed all the time? No. It doesn’t.
Once you get over trying to be “that other person,” the person you once were, being the best possible version of the person you are today ought to seem less daunting.