There is no built-in way to change the default behavior and the look and feel of tabs in the Google Sheets web app other than assigning a color to the tab, but there are a few options: All Sheets button, Name box, among others.
All Sheets button
This option might be preferred by users that like to handle sheets opening by point and click and look for no setup, no maintenance option. Clicking the All Sheets button will show a list of all the sheets. It might not be the best option when having a very large list of sheets and on small screens.
This option might be preferred by users who like to use keyboard shortcuts and type over multiple clicks and scrolling. On the *Name box you can type the sheet name. The autocomplete option will show the matching sheets' names as you type; it's possible to select the sheet from the autocomplete options using the keyboard.
Google Apps Script
You might extend Google Sheets by using Google Apps Script. You could create a custom menu and Editor Add-On or Workspace Add-On (a side panel).
From the custom menu, you might open a dialog, a sidebar or open specific sheets.
Non built-in features
Web browser bookmarks
Each sheet has its URL. You could use your web browser bookmarks, especially if you have a few sheets that you need to open frequently.
I haven't used userscripts with Google Workspace editors (Google Docs, Drawings, Forms, Sheets, Slides), but it might be possible; just be aware that Google might change the elements ID and the class names anytime, causing the userscript to break and require more maintenance than other options. Because of this, you might prefer using stable features and the Google Sheets API to a
userscript to play with the DOM.