Can an attorney serve papers on a party just like any other non-party?

Generally, yes. In California, an attorney may serve process or other papers so long as the attorney is at least 18 years old and not a party to the action. Cal. Civ. Proc. Code § 414.10 [1]

“A summons may be served by any person who is at least 18 years of age and not a party to the action.” Cal. Civ. Proc. Code § 414.10 [1]

For a federal action, the same basic rule applies: service may be made by any person who is at least 18 and not a party. Fed. R. Civ. P. 4(c)(2) [3]

Important distinction

• Serving process: You may personally serve a summons and complaint if you are not a named party.
• Serving papers after the action begins: You may generally serve pleadings, motions, notices, and other documents, but the applicable service method, timing, and proof-of-service requirements still control. For example, California mail service generally requires a proper proof of service under Cal. Civ. Proc. Code § 1013a [2].
• Your client’s case: You cannot serve papers as the attorney if you are yourself a named party or otherwise legally disqualified from serving that document.
• Family-law matters: Check the specific form, order, and local rule. Some family-law filings require personal service, service by someone other than the protected party, or a particular notice-and-acknowledgment procedure. An attorney’s status alone does not eliminate those requirements.

Thus, your being an attorney does not prevent service; your eligibility depends principally on being 18 or older, nonparty, and compliant with the governing service rules.