Greencat, the information you have given is irresponsible to say the least

Ebay and paypal do use collection agencies if and when necessary.

And ARL is definitely a registered collection agency with memberships of many of the collection industry associations.

The OP would be well advised to communicate with them, not delete and ignore the emails.

There could well be a scam involved if accounts have been opened in her Mother's name with ebay and paypal, but that needs to be sorted out with ebay and paypal first.....a collection agency only tries to collect what is owing to their clients.  They are not in the business of verifying whether those amounts are from legitimate accounts.