I assume if a Russian citizen were in the same boat, the residence permit affixed to their passport would suffice, but it's not an ID card.
AFAIK, no EU country will issue a proper ID card (valid throughout the EU) to non-citizens. In Spain at least you get a special foreigners' ID card that's more or less equal within Spain. In Germany the most you can hope for is your residence permit in card form, but I was not so lucky.
You can find specimens of residency permit cards (for non-EU citizens) here, which are typically sufficient ID to open a bank account:
https://www.consilium.europa.eu/prado/en/search-by-document-...
One of the first things I had to do upon arrival in Sweden was go to Migrationsverket to get my "Uppehållstillståndskort": a residence card. It has your residence status on it. I also had to get a Skatteverket "Identitetskort". Both of these have been accepted as valid identification documents both within and outside of Sweden, I've had to provide scans of one or the other numerous times.
I'm not sure about this but I think the German residence card, if you have one, is "ID" enough for a lot of things inside Germany, but there are still cases when you have to use your passport. I know at least one person who drives around with her residence permit and her driving license, but doesn't carry her passport regularly.
But I don't think that being able to travel with the thing is a requirement for it to be acceptable as an "additional photo ID" for the kinds of validation we're talking about here. Even if I can't travel with my Skatteverket or residence permit cards, they have been regularly accepted as ID verification for services based outside of Sweden. They've been sufficient when it comes to showing that I do indeed have the relevant Swedish identification documents and reside here. I think once, possibly for E-trade, this was coupled with having to provide a bill sent to my Swedish address. But that shouldn't really be a big issue either for anyone actually residing in another country.
To think that a couple of months ago I was hoping businesses wouldn't target Russians based on passport as an extra mile in enacting sanctions. Now the only options increasingly are seeking asylum or transferring my existing customers and moving to Russia to seek work there.