The existing panel and breaker should be fine.
The panel is rated for up to 100A, the maximum current it is configured for based on wire size and matching breaker feeding the panel is 50A. You could use a 200A panel out there being fed by #8 thwn, and you could still only feed it with a 50A breaker due to wire size.
You need to separate the grounds and neutrals. it appears (though I can't be certain) that the bus you have is a neutral bus and should be isolated from ground. The panel cover should list a part number for a ground bus, some inspectors will insist that specific UL/CSA/ETL "Listed" part be used, but not always.
Also because of the warm ambient temperature in a hot tub the instructions may say to feed the tub from the disconnect with a specified minimum wire size that would normally be considered oversized. The instructions are part of the "Listing" and they must be followed.
Finally note if you have to change the wire to the tub from the disconnect install an insulated ground wire like before, the code doesn't allow bare grounds for hot tubs.