What you're describing is Siri's ability to handle back-to-back requests without having to say “Siri” each time.
There isn't a separate setting to enable this feature. Normally, you can make another request immediately after the first one.
I'd first check whether it happens only with Spotify. Try asking Siri about the weather, then immediately ask a follow-up question without saying “Siri” again.
If that works, the issue may be related to how Siri handles Spotify commands. If Siri closes immediately in both cases, it points more toward Siri itself.
You can also check Settings > Accessibility > Siri > Siri Pause Time, although that setting controls how long Siri waits for you to finish speaking, not how long it listens for follow-up requests.
Does this happen with all Siri commands, or only when skipping songs in Spotify?
A friendly note: This is the German-language Apple Support Community. Posting your question in German will make it easier for more members to understand it and offer assistance.