I've never used a managed switch that had that behavior, and wouldn't find it acceptable. Normally I would expect either that you can configure the management interface to be on any VLAN or that the management interface would always be on the untagged network accessible via a trunk port. I'd say that either the switch is misconfigured or defective by design.
I think it's just missing that essential configuration setting, ie. which VLAN to use for the management interface. I even went so far as to RTFM which is
here.
I thought it was really odd not to have a simple setting for the management VLAN, so I even went so far as to check the manuals for some other competing switches. Zyxel GS1200-8 is very cheap indeed and supports setting the management VLAN:
https://download.zyxel.com/GS1200-8/user_guide/GS1200-8_V2.00_Ed2.pdfSo does D-Link DGS-1100-08V2/B
https://support.dlink.com/resource/products/DGS-1100-08V2/REVA/DGS-1100-08V2_REVA_MANUAL_v1.00_WW.pdfMy other 8 port managed switches are Netgear, GS108E and GS308E. They seem to default to VLAN 1 for management; it can't be set manually.
I also have a TP-link TL-SG1024DE which also seems to pick VLAN 1 even though it's clearly running the same basic firmware as the 608E. Maybe I just got lucky as it's plugged directly into the router and it's on a UPS so very rarely gets reset.