My Mother's caretaker is from an agency and calls my mother MOM. Is this professional and appropriate?

Have you spoken to the caregiver?  First do that.  If you have done that, complain to the agency immediately.  I suspect the caregiver is trying to groom your mom to make significant gifts to her.

This is an attitude born out of dysfunctional families and the rejection experienced by many, wherein it is decided to choose one's own family. It needs correction if taken too far.

The line is drawn when sentiments are they are "Like a family", or
"better than one's own family". That can be sentimental but is usually understood by all it is just a sentiment. Not a fact. Or, people will say: "I am adopting you as my family".

When the real family is offended or concerned, or the sentiment is taken too far, put a stop to it. The caregiver could be delusional or just trying to find a place to belong. Lacks appropriate boundaries.

What you do is talk to the agency, demanding the caregiver be more professional. When she states this in front of you, correct her in a strong way. "She is not your Mom, you are not my sister".
You could make light of it, to say: "If that were true, we would not be paying you".

That is beyond presumptuous. And if your Mom is the least bit confused already, having someone other than you call her Mom could get extremely confusing for her. The CG may feel that your Mom will react better with this “family” atmosphere, but it is too much. There was one lady at Moms NH who was very far along in her dementia, who the nurse called Mom, but it was with the real daughters blessing, as it was the only way they could get any cooperation from her. But yours is not the same and I would stop it. You could tell the CG that your Mom was confused about who all her children are, and so you’d appreciate it if she’s call her Miss XYZ, or some term of endearment. If she gives you any guff over it, you know she’s not looking out for your Moms best interest.

I’m going to give the caregiver the benefit of the doubt and assume she doesn’t understand professional boundaries and likely believes she is doing a great job by endearing herself to your mother. Since you otherwise like her work, I would gently but firmly explain your expectations to her regarding this. If it doesn’t change, then I would discuss it with the agency. See how it goes. Moving forward, if you still have a bad feeling about her, she needs to go. Listen to your gut.

There were some women at my mom's nursing home who did this to several of the residents, and it always annoyed the heck out of me. If you have told her point blank to stop and she won't then I'm not sure what you can do besides look for a replacement, IMO anyone who thinks her brief acquaintance with your mother is equal to or even supersedes your own relationship with her is beyond creepy.

I find this sort of creepy too. Our CG for my dad loved my parents, always said they were like her own grandparents,, but she NEVER called them "Grandpa or ma".

This is not appropriate and should be stopped.
Either you talk to the caregiver or contact the agency and they should have a talk with the caregiver.
I do hope the caregiver does not have access to any accounts, credit cards or any other private sensitive information. And I hope your mom does not have access to credit cards, cash, checkbook or any other financial information. I also hope there are no valuables in the house that mom could give to her "daughter"

Inappropriate. Creepy. Suspicious...

I would contact the agency right now. Fire her!

this is creepy. And, she might very well be setting you Mom up for a scam.....sure sounds like it. Calling you sister...and making that connection to you with your Mom. Oh heck no...fire her.

get rid of her now.

This is highly inappropriate. I understand how you don’t want to rock the boat with mom. At the same time you must put an end to this behavior. I recommend talking to CG first. Start by emphasizing the positives of how her care and how you appreciate her. Then spell out exactly how you want CG to address your mom, you and all other family members. Explain that this is mandatory. Evaluate her response. Was she understanding, apologetic, respectful of you and willing to cooperate? If yes, give her another chance. If no, call the agency and fire her. There are other good CGs out there.

It reminds me of a run in I had with my mom’s house long time house cleaner who claimed to know my mom better than I did and thought she could tell me how to care for my mother. My mom absolutely adored her. To make a very long story short, house cleaner has been fired and life is more peaceful. Turns out she was a poison pill trying to sabotage my relationship with my mom.