Yes - your mum can still give gifts, even if she is self-funding. However, it is important in that case that the gifts be "reasonable" in value - a guide that is often used is to look at the type of gift that your mum had regularly given in the past. Otherwise, suddenly giving a lot more might be construed as a deprivations of asset - ie, giving away money or items so as to exclude their being used to pay for care.
Attorneys can also give gifts on the donor's behalf - even to themselves - again, the guide is that they must be "reasonable" and are what the donor themselves would have given, were they able to do so. As before, the sort of gifts given in previous years would be a good guide to this.
Of course, attorneys need to be particularly careful when giving to themselves, for they have a duty not to use their authority for their own benefit.
For example: if your mum had been in the custom of giving someone £25 for birthday and Christmas, there is no reason as to why this shold not continue. But it would be most unwise to start giving substantially more, or outside of special occasions. That would be suspiscious.
Also, if someone is in care and self-funding, they do retain some of their income (about £20 a week as I recall). This may be used for any purpose they wish - the giving of gifts is an obvious one.
A "reasonable" gift would, therefore, usually be a continuance of one customarily given, it wuld normally be on a special occasion and as the guidance states, of a value appropriate to the person's resources.